This is GroebnerStrata.info, produced by Macaulay2, version 1.25.05

GroebnerStrata -- a package for creating families of ideals with the same initial ideal
***************************************************************************************

Description
===========

The GroebnerStrata package is designed to compute, given a monomial ideal in a
polynomial ring, (and a term order, coming from the polynomial ring), a family
of homogeneous ideals all with the given monomial ideal as its lead term
(initial) ideal.

The GroebnerStrata package is designed for computing homogeneous strata and
parameter families for monomial ideals in polynomial rings. In certain
instances homogeneous strata can be used to compute local coordinates on
Hilbert Schemes.

Here is an example of the basic use of the package.  We compute the Groebner
family of the ideal $(a^2, ab, b^2, ac)$.  This is an ideal in a polynomial
ring with variables the same as $S$, and whose coefficient ring is a polynomial
ring containing all of the parameters.  The groebnerStratum(Ideal),
function returns the ideal in the parameters of the locus of
parameters, for which the given family is a Groebner basis.

+-----------------------------------------------------------------------------+
|i1 : kk = ZZ/101;                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = kk[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i3 : M = ideal"a2,ab,b2,ac";                                                 |
|                                                                             |
|o3 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i4 : F = groebnerFamily M;                                                   |
|                                                                             |
|o4 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  ,   |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17    |
|-----------------------------------------------------------------------------|
|t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]                   |
| 23   7   9   14   16   20   22   13   15   19   21                          |
+-----------------------------------------------------------------------------+
|i5 : netList F_*                                                             |
|                                                                             |
|     +-------------------------------------------------------+               |
|     | 2                      2                      2       |               |
|o5 = |a  + t b*c + t a*d + t c  + t b*d + t c*d + t d        |               |
|     |      1       3       2      4       5       6         |               |
|     +-------------------------------------------------------+               |
|     |                         2                         2   |               |
|     |a*b + t b*c + t a*d + t c  + t  b*d + t  c*d + t  d    |               |
|     |       7       9       8      10       11       12     |               |
|     +-------------------------------------------------------+               |
|     | 2                         2                         2 |               |
|     |b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d  |               |
|     |      13       15       14      16       17       18   |               |
|     +-------------------------------------------------------+               |
|     |                            2                         2|               |
|     |a*c + t  b*c + t  a*d + t  c  + t  b*d + t  c*d + t  d |               |
|     |       19       21       20      22       23       24  |               |
|     +-------------------------------------------------------+               |
+-----------------------------------------------------------------------------+
|i6 : J = trim groebnerStratum F                                              |
|                                                                             |
|                                                                             |
|o6 = ideal (t  - t   + t  t  , t  t   - t  t  t  , t   - t   + t  t   - t t  |
|             7    20    13 19   22 15    15 19 21   10    23    22 13    9 19|
|     ------------------------------------------------------------------------|
|                    2                                                        |
|     + t  t   - t  t   + t  t   - t  t  t  , t  + t  t  , t  + 2t  t   -     |
|        16 19    15 19    20 21    13 19 21   8    14 19   1     20 19       |
|     ------------------------------------------------------------------------|
|         2                                                                   |
|     t  t  , t t   + 50t  t   - 50t  t  t   - t t  t   - 50t  t  t   +       |
|      13 19   9 22      16 22      22 13 21    9 19 21      16 19 21         |
|     ------------------------------------------------------------------------|
|              2                                                              |
|     50t  t  t  , t   + 50t  t   - t  t   - 50t  t  t   - 50t  t  t   +      |
|        13 19 21   24      16 22    23 21      22 13 21      16 19 21        |
|     ------------------------------------------------------------------------|
|         2             2          2                                          |
|     t  t   + 50t  t  t  , t   + t  - t t   - t t   + t  t   + t t  t   -    |
|      20 21      13 19 21   18    9    9 16    3 15    23 15    9 15 19      |
|     ------------------------------------------------------------------------|
|                  2  2                                                       |
|     t  t  t   + t  t   - t  t   + t t  t   - 2t  t  t   + t  t  t  t   +    |
|      16 15 19    15 19    17 21    9 13 21     20 15 21    13 15 19 21      |
|     ------------------------------------------------------------------------|
|         2                                                                   |
|     t  t  , t   - t t   + t  t   + t  t   - t  t  t   - t  t  t  , t  +     |
|      14 21   11    9 20    14 22    17 19    20 15 19    14 19 21   4       |
|     ------------------------------------------------------------------------|
|                                                  2        3                 |
|     2t  t   - t t   + 2t  t   - 2t  t  t   - t  t   + t  t   - 4t  t  t   + |
|       20 22    3 19     23 19     22 13 19    16 19    15 19     20 19 21   |
|     ------------------------------------------------------------------------|
|          2            2        2                                            |
|     2t  t  t  , t  + t   - t  t  , t  t   + 50t  t  t   - 2t  t  t   -      |
|       13 19 21   2    20    14 19   17 22      16 22 13     14 22 21        |
|     ------------------------------------------------------------------------|
|           2                                            2       2     2      |
|     50t  t  t   - t  t  t   - 50t  t  t  t   + 2t  t  t   + 50t  t  t  ,    |
|        22 13 21    17 19 21      16 13 19 21     14 19 21      13 19 21     |
|     ------------------------------------------------------------------------|
|                        2                                                    |
|     t t   - 2t  t   + t  t   + t  t  t   + 4t  t  t   - t t  t   +          |
|      3 22     23 22    22 13    16 22 19     20 22 21    3 19 21            |
|     ------------------------------------------------------------------------|
|                                      2               2         2  2         |
|     2t  t  t   - 3t  t  t  t   - t  t  t   - 4t  t  t   + 2t  t  t  , t   - |
|       23 19 21     22 13 19 21    16 19 21     20 19 21     13 19 21   12   |
|     ------------------------------------------------------------------------|
|                            2                                                |
|     t  t  - 50t  t  t   - t t   + t t  t   + t t  t   - 2t  t  t   -        |
|      23 9      16 22 13    9 19    9 16 19    3 15 19     23 15 19          |
|     ------------------------------------------------------------------------|
|           2           2     2  3                                 2          |
|     t t  t   + t  t  t   - t  t   + t t  t   + t  t  t   + 50t  t  t   +    |
|      9 15 19    16 15 19    15 19    9 20 21    14 22 21      22 13 21      |
|     ------------------------------------------------------------------------|
|                                                                       2     |
|     t  t  t   - t t  t  t   + 50t  t  t  t   + 3t  t  t  t   - t  t  t  t   |
|      17 19 21    9 13 19 21      16 13 19 21     20 15 19 21    13 15 19 21 |
|     ------------------------------------------------------------------------|
|               2       2     2                                           2   |
|     - 2t  t  t   - 50t  t  t  , t  - t t   + 2t  t   - 2t  t  t   - t  t   +|
|         14 19 21      13 19 21   5    3 20     23 20     14 22 19    17 19  |
|     ------------------------------------------------------------------------|
|            2      2            2                    2                    2  |
|     t  t  t   - 2t  t   + 2t  t  t  , t  - t t   + t   + t  t  t   - t  t   |
|      20 15 19     20 21     14 19 21   6    3 23    23    16 20 22    14 22 |
|     ------------------------------------------------------------------------|
|                       2 2          2          2            2          3     |
|     - t  t  t  t   + t t   - t t  t   - t t  t   + 2t  t  t   + t t  t   -  |
|        16 22 13 19    9 19    9 16 19    3 15 19     23 15 19    9 15 19    |
|     ------------------------------------------------------------------------|
|            3     2  4                                                       |
|     t  t  t   + t  t   + t t  t   - 2t  t  t   - t  t  t  t   - t  t  t  t  |
|      16 15 19    15 19    3 20 21     23 20 21    20 22 13 21    16 20 19 21|
|     ------------------------------------------------------------------------|
|           2              2             2              2               2     |
|     + t  t  t  t   - t  t  t   + t t  t  t   + t  t  t  t   - 3t  t  t  t   |
|        22 13 19 21    17 19 21    9 13 19 21    16 13 19 21     20 15 19 21 |
|     ------------------------------------------------------------------------|
|              3        2  2              2         2  2     2  2  2          |
|     + t  t  t  t   + t  t   + t  t  t  t   + 2t  t  t   - t  t  t  )        |
|        13 15 19 21    20 21    20 13 19 21     14 19 21    13 19 21         |
|                                                                             |
|o6 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17   2|
|-----------------------------------------------------------------------------|
| , t , t , t  , t  , t  , t  , t  , t  , t  , t  ]                           |
|3   7   9   14   16   20   22   13   15   19   21                            |
+-----------------------------------------------------------------------------+

The ideal of the parameter space of all homogeneous ideals with this lead term
ideal is an ideal in 24 variables.  Often, these parameter ideals are in too
many variables to easily analyze them.  But in this case we can determine the
irreducible components of the ideal $J$.  There are two components, of
dimensions 8 and 11.  Note that they are both rational varieties.

+-----------------------------------------------------------------------------+
|i7 : compsJ = decompose J;                                                   |
+-----------------------------------------------------------------------------+
|i8 : compsJ = compsJ/trim;                                                   |
+-----------------------------------------------------------------------------+
|i9 : compsJ/dim                                                              |
|                                                                             |
|o9 = {11, 8}                                                                 |
|                                                                             |
|o9 : List                                                                    |
+-----------------------------------------------------------------------------+
|i10 : netList compsJ_0_*                                                     |
|                                                                             |
|      +----------------------------------------------------------------------|
|o10 = |t   - t  t                                                            |
|      | 22    19 21                                                          |
|      +----------------------------------------------------------------------|
|      |t  - t   + t  t                                                       |
|      | 7    20    13 19                                                     |
|      +----------------------------------------------------------------------|
|      |                                 2                                    |
|      |t   - t   - t t   + t  t   - t  t   + t  t                            |
|      | 10    23    9 19    16 19    15 19    20 21                          |
|      +----------------------------------------------------------------------|
|      |t  + t  t                                                             |
|      | 8    14 19                                                           |
|      +----------------------------------------------------------------------|
|      |                   2                                                  |
|      |t  + 2t  t   - t  t                                                   |
|      | 1     20 19    13 19                                                 |
|      +----------------------------------------------------------------------|
|      |                   2                                                  |
|      |t   - t  t   + t  t                                                   |
|      | 24    23 21    20 21                                                 |
|      +----------------------------------------------------------------------|
|      |       2                                                    2  2      |
|      |t   + t  - t t   - t t   + t  t   + t t  t   - t  t  t   + t  t   - t |
|      | 18    9    9 16    3 15    23 15    9 15 19    16 15 19    15 19    1|
|      +----------------------------------------------------------------------|
|      |t   - t t   + t  t   - t  t  t                                        |
|      | 11    9 20    17 19    20 15 19                                      |
|      +----------------------------------------------------------------------|
|      |                           2        3                                 |
|      |t  - t t   + 2t  t   - t  t   + t  t   - 2t  t  t                     |
|      | 4    3 19     23 19    16 19    15 19     20 19 21                   |
|      +----------------------------------------------------------------------|
|      |      2        2                                                      |
|      |t  + t   - t  t                                                       |
|      | 2    20    14 19                                                     |
|      +----------------------------------------------------------------------|
|      |               2                                               2      |
|      |t   - t  t  - t t   + t t  t   + t t  t   - 2t  t  t   - t t  t   + t |
|      | 12    23 9    9 19    9 16 19    3 15 19     23 15 19    9 15 19    1|
|      +----------------------------------------------------------------------|
|      |                           2           2      2                       |
|      |t  - t t   + 2t  t   - t  t   + t  t  t   - 2t  t                     |
|      | 5    3 20     23 20    17 19    20 15 19     20 21                   |
|      +----------------------------------------------------------------------|
|      |              2     2 2          2          2            2          3 |
|      |t  - t t   + t   + t t   - t t  t   - t t  t   + 2t  t  t   + t t  t  |
|      | 6    3 23    23    9 19    9 16 19    3 15 19     23 15 19    9 15 19|
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                  2                          |
| t   + t t  t   - 2t  t  t   + t  t  t  t   + t  t                           |
|7 21    9 13 21     20 15 21    13 15 19 21    14 21                         |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|     2     2  3                                                              |
| t  t   - t  t   + t t  t   + t  t  t   - t t  t  t   + 3t  t  t  t   - t  t |
|6 15 19    15 19    9 20 21    17 19 21    9 13 19 21     20 15 19 21    13 1|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|          3     2  4                                2             2          |
| - t  t  t   + t  t   + t t  t   - 2t  t  t   - t  t  t   + t t  t  t   - 3t |
|    16 15 19    15 19    3 20 21     23 20 21    17 19 21    9 13 19 21     2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|  2              2                            |                              |
| t  t   - t  t  t                             |                              |
|5 19 21    14 19 21                           |                              |
|----------------------------------------------+                              |
|                                              |                              |
|                                              |                              |
|                                              |                              |
|----------------------------------------------+                              |
|     2              3        2  2        2  2 |                              |
| t  t  t   + t  t  t  t   + t  t   + t  t  t  |                              |
|0 15 19 21    13 15 19 21    20 21    14 19 21|                              |
|----------------------------------------------+                              |
+-----------------------------------------------------------------------------+
|i11 : netList compsJ_1_*                                                     |
|                                                                             |
|      +----------------------------------------------------------------------|
|o11 = |t                                                                     |
|      | 15                                                                   |
|      +----------------------------------------------------------------------|
|      |t  + 50t   - 50t  t                                                   |
|      | 9      16      13 21                                                 |
|      +----------------------------------------------------------------------|
|      |t  - t   + t  t                                                       |
|      | 7    20    13 19                                                     |
|      +----------------------------------------------------------------------|
|      |                              2                                       |
|      |t   + 50t  t   - 2t  t   - 50t  t                                     |
|      | 17      16 13     14 21      13 21                                   |
|      +----------------------------------------------------------------------|
|      |t   - t   + t  t   - 50t  t   + t  t   + 50t  t  t                    |
|      | 10    23    22 13      16 19    20 21      13 19 21                  |
|      +----------------------------------------------------------------------|
|      |t  + t  t                                                             |
|      | 8    14 19                                                           |
|      +----------------------------------------------------------------------|
|      |t  - 2t   + t  t   + t  t   + 4t  t   - 2t  t  t                      |
|      | 3     23    22 13    16 19     20 21     13 19 21                    |
|      +----------------------------------------------------------------------|
|      |                   2                                                  |
|      |t  + 2t  t   - t  t                                                   |
|      | 1     20 19    13 19                                                 |
|      +----------------------------------------------------------------------|
|      |                                                          2           |
|      |t   + 50t  t   - t  t   - 50t  t  t   - 50t  t  t   + t  t   + 50t  t |
|      | 24      16 22    23 21      22 13 21      16 19 21    20 21      13 1|
|      +----------------------------------------------------------------------|
|      |         2        2       2  2                                        |
|      |t   + 25t   - t  t   - 25t  t                                         |
|      | 18      16    14 21      13 21                                       |
|      +----------------------------------------------------------------------|
|      |                                                                     2|
|      |t   + 50t  t   + t  t   - 50t  t  t   - 50t  t  t   + t  t  t   + 50t |
|      | 11      16 20    14 22      16 13 19      20 13 21    14 19 21      1|
|      +----------------------------------------------------------------------|
|      |t  + 2t  t   - t  t  t                                                |
|      | 4     20 22    22 13 19                                              |
|      +----------------------------------------------------------------------|
|      |      2        2                                                      |
|      |t  + t   - t  t                                                       |
|      | 2    20    14 19                                                     |
|      +----------------------------------------------------------------------|
|      |                                  2                                   |
|      |t   + 50t  t   - 50t  t  t   - 25t  t   - 50t  t  t   + t  t  t   - 50|
|      | 12      23 16      16 22 13      16 19      16 20 21    14 22 21     |
|      +----------------------------------------------------------------------|
|      |                                                   2      2           |
|      |t  + t  t  t   + t  t  t   - 2t  t  t   + 50t  t  t   + 2t  t   - 2t  |
|      | 5    20 22 13    16 20 19     14 22 19      16 13 19     20 21     20|
|      +----------------------------------------------------------------------|
|      |      2                    2                                          |
|      |t  - t   + t  t  t   - t  t   + t  t  t   + t  t  t   - t  t  t  t   +|
|      | 6    23    16 20 22    14 22    23 22 13    23 16 19    16 22 13 19  |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|  2                                                                          |
| t                                                                           |
|9 21                                                                         |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
| t  t                                                                        |
|3 19 21                                                                      |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                  2                                 2       2     2          |
|t  t  t   + 50t  t  t   + 50t  t  t  t   + 50t  t  t   - 25t  t  t           |
| 23 13 21      22 13 21      16 13 19 21      20 13 21      13 19 21         |
|-----------------------------------------------------------------------------|
|               2  2                                                          |
|t  t  t   - 50t  t  t                                                        |
| 13 19 21      13 19 21                                                      |
|-----------------------------------------------------------------------------|
|    2  2                                                                     |
| 25t  t   + 4t  t  t   - 2t  t  t  t   - 2t  t  t  t   - 2t  t  t  t   + t  t|
|    16 19     23 20 21     20 22 13 21     16 20 19 21     23 13 19 21    22 |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|                                                               |             |
|                                                               |             |
|                                                               |             |
|---------------------------------------------------------------+             |
|2                 2         2  2               2       2  2  2 |             |
|  t  t   + t  t  t  t   - 3t  t   + 3t  t  t  t   - 26t  t  t  |             |
|13 19 21    16 13 19 21     20 21     20 13 19 21      13 19 21|             |
|---------------------------------------------------------------+             |
+-----------------------------------------------------------------------------+

This tells us that there are 2 components (at least over the given field).
Their dimensions are 11, 8.

We can find random points on each component, since these components are
rational.

+-----------------------------------------------------------------------------+
|i12 : pt1 = randomPointOnRationalVariety compsJ_0                            |
|                                                                             |
|o12 = | 7 -19 15 -31 26 37 25 25 38 -29 -23 34 19 -10 6 -29 -36 -30 -8 -49   |
|      -----------------------------------------------------------------------|
|      -22 -29 19 24 |                                                        |
|                                                                             |
|               1       24                                                    |
|o12 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i13 : pt2 = randomPointOnRationalVariety compsJ_1                            |
|                                                                             |
|o13 = | -13 -8 44 30 39 49 46 -41 -5 -12 -33 0 24 21 49 34 -38 -16 34 19 -47 |
|      -----------------------------------------------------------------------|
|      0 39 -24 |                                                             |
|                                                                             |
|               1       24                                                    |
|o13 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i14 : F1 = sub(F, (vars S)|pt1)                                              |
|                                                                             |
|              2              2                             2                 |
|o14 = ideal (a  + 38b*c - 31c  - 29a*d + 26b*d - 19c*d + 7d , a*b + 6b*c -   |
|      -----------------------------------------------------------------------|
|         2                              2   2              2                 |
|      23c  - 29a*d + 34b*d + 37c*d + 15d , b  - 22b*c - 36c  - 29a*d - 30b*d |
|      -----------------------------------------------------------------------|
|                   2                  2                              2       |
|      + 19c*d + 25d , a*c + 19b*c - 8c  + 24a*d - 49b*d - 10c*d + 25d )      |
|                                                                             |
|o14 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i15 : F2 = sub(F, (vars S)|pt2)                                              |
|                                                                             |
|              2             2                             2                  |
|o15 = ideal (a  - 5b*c + 30c  - 12a*d + 39b*d - 8c*d - 13d , a*b + 49b*c -   |
|      -----------------------------------------------------------------------|
|         2                      2   2              2                      2  |
|      33c  + 34a*d + 49c*d + 44d , b  - 47b*c - 38c  - 16b*d + 24c*d + 46d , |
|      -----------------------------------------------------------------------|
|                       2                              2                      |
|      a*c + 39b*c + 34c  - 24a*d + 19b*d + 21c*d - 41d )                     |
|                                                                             |
|o15 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i16 : decompose F1                                                           |
|                                                                             |
|                                   2              2                      2   |
|o16 = {ideal (a + 19b - 8c - 20d, b  - 22b*c - 36c  + 16b*d - 11c*d - 50d ), |
|      -----------------------------------------------------------------------|
|      ideal (c + 24d, b + 26d, a - 45d)}                                     |
|                                                                             |
|o16 : List                                                                   |
+-----------------------------------------------------------------------------+
|i17 : decompose F2                                                           |
|                                                                             |
|                               2                              2   2          |
|o17 = {ideal (a*c + 39b*c + 34c  - 24a*d + 19b*d + 21c*d - 41d , b  - 47b*c -|
|      -----------------------------------------------------------------------|
|         2                      2                   2                      2 |
|      38c  - 16b*d + 24c*d + 46d , a*b + 49b*c - 33c  + 34a*d + 49c*d + 44d ,|
|      -----------------------------------------------------------------------|
|       2             2                             2                         |
|      a  - 5b*c + 30c  - 12a*d + 39b*d - 8c*d - 13d )}                       |
|                                                                             |
|o17 : List                                                                   |
+-----------------------------------------------------------------------------+

Note, the general element of one component is a plane conic union a point. (The
dimension of the locus of all such is: (choice of plane) + (choice of degree 2
in plane) + choice of point. This is 3 + 5 + 3 = 11.

The other component consists of two skew lines.  This has dimension (choice of
line) + (choice of line). This is 4 + 4 = 8.  Also notice that the 2 skew lines
do not have to be defined over the base field, as in this case.

Authors
=======

  * Mike Stillman (see http://www.math.cornell.edu/~mike/ )
    <mike@math.cornell.edu>
  * Kristine Jones <kejones84@gmail.com>

Version
=======

This documentation describes version 0.9 of GroebnerStrata.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{GroebnerStrataSource,                                                  |
|  title = {{GroebnerStrata: computing Groebner loci in                       |
|  author = {Mike Stillman and Kristine Jones},                               |
|  howpublished = {A \emph{Macaulay2} package available                       |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2                      |
|}                                                                            |
|-----------------------------------------------------------------------------|
|Hilbert schemes. Version~0.9}},                                              |
|                                                                             |
|at                                                                           |
|/Macaulay2/packages}}                                                        |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * findWeightConstraints -- returns a matrix
        of weight constraints
      * findWeightVector -- returns a weight vector
      * groebnerFamily -- computes families of ideals
        with a specified initial ideal
      * groebnerStratum -- compute the ideal where a
        given is a Groebner basis
      * linearPart -- returns linear part of a polynomial
      * nonminimalMaps -- find the degree zero maps in
        the Schreyer resolution of an ideal
      * "randomPointOnRationalVariety" -- see
        randomPointOnRationalVariety(Ideal),
        -- find a random point on a
        variety that can be detected to be rational
      * "randomPointsOnRationalVariety" -- see
        randomPointsOnRationalVariety(Ideal,ZZ),
        -- find random points on
        a variety that can be detected to be rational
      * smallerMonomials -- returns the standard
        monomials smaller but of the same degree as given monomial(s)
      * standardMonomials -- computes standard
        monomials
      * tailMonomials -- find tail monomials
  * Methods
      * "findWeightConstraints(Ideal,List)" -- see findWeightConstraints,
        -- returns a matrix of weight constraints
      * "findWeightVector(Ideal,List)" -- see findWeightVector,
        -- returns a weight vector
      * "groebnerFamily(Ideal)" -- see groebnerFamily, --
        computes families of ideals with a specified initial ideal
      * "groebnerFamily(Ideal,List)" -- see groebnerFamily,
        -- computes families of ideals with a specified initial
        ideal
      * "groebnerStratum(Ideal)" -- see groebnerStratum,
        -- compute the ideal where a given is a Groebner basis
      * "nonminimalMaps(Ideal)" -- see nonminimalMaps, --
        find the degree zero maps in the Schreyer resolution of an ideal
      * randomPointOnRationalVariety(Ideal)
        -- find a random point on a
        variety that can be detected to be rational
      * randomPointsOnRationalVariety(Ideal,ZZ)
        -- find random points on
        a variety that can be detected to be rational
      * "smallerMonomials(Ideal)" -- see smallerMonomials,
        -- returns the standard monomials smaller but of the
        same degree as given monomial(s)
      * "smallerMonomials(Ideal,RingElement)" -- see smallerMonomials,
        -- returns the standard monomials smaller but of the
        same degree as given monomial(s)
      * "standardMonomials(Ideal)" -- see standardMonomials,
        -- computes standard monomials
      * "standardMonomials(List,Ideal)" -- see standardMonomials,
        -- computes standard monomials
      * "standardMonomials(ZZ,Ideal)" -- see standardMonomials,
        -- computes standard monomials
      * "tailMonomials(Ideal)" -- see tailMonomials, --
        find tail monomials
      * "tailMonomials(Ideal,RingElement)" -- see tailMonomials,
        -- find tail monomials
  * Symbols
      * AllStandard -- boolean option for determining the
        use of all standard or smaller standard monomials
      * Minimalize -- boolean option for determining whether
        excess parameters will be eliminated

For the programmer
==================

The object GroebnerStrata, is a package (Macaulay2Doc),
defined in GroebnerStrata.m2.

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:441:0.

AllStandard -- boolean option for determining the use of all standard or smaller standard monomials
***************************************************************************************************

Description
===========

This is an optional input for the groebnerStratum, and
groebnerFamily, functions.  It takes values $true$ and
$false$.  When assigned the value $true$, the functions are computed with
respect to all standard monomials for the specified generators.  When assigned
the value $false$, the functions are computed with respect to smaller monomials
of the same degree as the specified generators but which do not lie in the
ideal.

See also
========

  * groebnerStratum -- compute the ideal where a given
    is a Groebner basis
  * groebnerFamily -- computes families of ideals with a
    specified initial ideal
  * smallerMonomials -- returns the standard monomials
    smaller but of the same degree as given monomial(s)
  * standardMonomials -- computes standard monomials

Functions with optional argument named AllStandard:
===================================================

  * "groebnerFamily(...,AllStandard=>...)" -- see groebnerFamily,
    -- computes families of ideals with a specified initial
    ideal
  * "tailMonomials(...,AllStandard=>...)" -- see tailMonomials,
    -- find tail monomials

For the programmer
==================

The object AllStandard, is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:1045:0.

findWeightConstraints -- returns a matrix of weight constraints
***************************************************************

  * Usage:
        constraints = findWeightConstraints(M, L)
  * Inputs:
      * M, an ideal (Macaulay2Doc), M should be a monomial ideal
      * L, a list (Macaulay2Doc), a list of lists of tail monomials
  * Outputs:
      * a matrix (Macaulay2Doc), with the constraints on a weight
        vector for $M$ to be the lead monomial considering the given tail
        monomials in $L$. The number of rows of the matrix is the number of
        variables in the ring of $M$, The number of columns is the total size
        of $L$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a,b,c, d];                                                 |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, a*b, b^2);                                              |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : L = smallerMonomials M;                                                 |
+-----------------------------------------------------------------------------+
|i4 : mat = findWeightConstraints(M,L)                                        |
|                                                                             |
|o4 = | 1  2  2  1  2  2  2  0  1  1  0  1  1  1  -1 0  0  -1 0  0  0  |      |
|     | 0  -1 0  0  -1 0  0  1  0  1  1  0  1  1  2  1  2  2  1  2  2  |      |
|     | -1 -1 -2 0  0  -1 0  -1 -1 -2 0  0  -1 0  -1 -1 -2 0  0  -1 0  |      |
|     | 0  0  0  -1 -1 -1 -2 0  0  0  -1 -1 -1 -2 0  0  0  -1 -1 -1 -2 |      |
|                                                                             |
|              4       21                                                     |
|o4 : Matrix ZZ  <-- ZZ                                                       |
+-----------------------------------------------------------------------------+
|i5 : needsPackage "Polyhedra"                                                |
|                                                                             |
|o5 = Polyhedra                                                               |
|                                                                             |
|o5 : Package                                                                 |
+-----------------------------------------------------------------------------+
|i6 : needsPackage "FourTiTwo"                                                |
|                                                                             |
|o6 = FourTiTwo                                                               |
|                                                                             |
|o6 : Package                                                                 |
+-----------------------------------------------------------------------------+
|i7 : dualCone posHull (-mat)                                                 |
|                                                                             |
|o7 = Cone{...2...}                                                           |
|                                                                             |
|o7 : Cone                                                                    |
+-----------------------------------------------------------------------------+
|i8 : rays oo                                                                 |
|                                                                             |
|o8 = | 0 0 0  0 |                                                            |
|     | 0 0 -1 1 |                                                            |
|     | 1 0 1  2 |                                                            |
|     | 0 1 1  2 |                                                            |
|                                                                             |
|              4       4                                                      |
|o8 : Matrix ZZ  <-- ZZ                                                       |
+-----------------------------------------------------------------------------+
|i9 : posHull mat -- seems wrong?                                             |
|                                                                             |
|o9 = Cone{...1...}                                                           |
|                                                                             |
|o9 : Cone                                                                    |
+-----------------------------------------------------------------------------+
|i10 : coneFromHData transpose mat                                            |
|                                                                             |
|o10 = Cone{...1...}                                                          |
|                                                                             |
|o10 : Cone                                                                   |
+-----------------------------------------------------------------------------+
|i11 : rays mat                                                               |
|                                                                             |
|o11 = 0                                                                      |
|                                                                             |
|               1                                                             |
|o11 : Matrix ZZ  <-- 0                                                       |
+-----------------------------------------------------------------------------+
|i12 : findWeightVector(M,L)                                                  |
|                                                                             |
|o12 = ({2, 2, 1, 1}, {1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1,  |
|      -----------------------------------------------------------------------|
|      1, 2, 2})                                                              |
|                                                                             |
|o12 : Sequence                                                               |
+-----------------------------------------------------------------------------+

TO BE FINISHED!! Note that the first generator listed for $M$ is $a^2$, and the
first corresponding standard monomial is $a*c$.  The difference of these two
monomials exponent vectors is $(1,0,-1,0)$.  This vector dotted with the weight
vector $(2,2,1,1)$ gives the value $1$, which is the first value in the second
list.

Note that the desired term ordering, and hence weight vector, may not exist. In
this case, null is returned.

+-------------------------------------------------+
|i13 : M = ideal"ab"                              |
|                                                 |
|o13 = ideal(a*b)                                 |
|                                                 |
|o13 : Ideal of R                                 |
+-------------------------------------------------+
|i14 : L1 = standardMonomials M                   |
|                                                 |
|         2   2             2                  2  |
|o14 = {{a , b , a*c, b*c, c , a*d, b*d, c*d, d }}|
|                                                 |
|o14 : List                                       |
+-------------------------------------------------+
|i15 : findWeightVector(M,L1)                     |
+-------------------------------------------------+

This command is used in the groebnerFamily, routine.

See also
========

  * groebnerStratum -- compute the ideal where a given
    is a Groebner basis
  * groebnerFamily -- computes families of ideals with a
    specified initial ideal

Ways to use findWeightConstraints:
==================================

  * "findWeightConstraints(Ideal,List)"

For the programmer
==================

The object findWeightConstraints, is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:673:0.

findWeightVector -- returns a weight vector
*******************************************

  * Usage:
        (w, h) = findWeightVector (M, L)
  * Inputs:
      * M, an ideal (Macaulay2Doc), M should be a monomial ideal
      * L, a list (Macaulay2Doc), a list of lists of standard
        monomials
  * Outputs:
      * w, a list (Macaulay2Doc), w is a weight vector that places
        the specified generators of M greater than the corresponding standard
        monomials, if possible.  If not possible, null is returned instead of
        (w,h)
      * h, a list (Macaulay2Doc), h is a list of values for w
        dotted with the difference of the exponent of the each standard
        monomial for each generator and the corresponding generator, in the
        order they are listed in L

Description
===========

In the first entry, this command returns a weight vector associated to a
monomial order that places the generators of a monomial ideal $M$ ahead of
standard monomials of the same degree.  The second entry is a list of values
for the weight vector dotted with the difference of the exponent of each
standard monomial for each generator and the corresponding generator.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a,b,c, d];                                                 |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, a*b, b^2);                                              |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : L = smallerMonomials M;                                                 |
+-----------------------------------------------------------------------------+
|i4 : findWeightVector(M,L)                                                   |
|                                                                             |
|o4 = ({2, 2, 1, 1}, {1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1,|
|     ------------------------------------------------------------------------|
|     2, 2})                                                                  |
|                                                                             |
|o4 : Sequence                                                                |
+-----------------------------------------------------------------------------+

Note that the first generator listed for $M$ is $a^2$, and the first
corresponding standard monomial is $a*c$.  The difference of these two
monomials exponent vectors is $(1,0,-1,0)$.  This vector dotted with the weight
vector $(2,2,1,1)$ gives the value $1$, which is the first value in the second
list.

Note that the desired term ordering, and hence weight vector, may not exist. In
this case, null is returned.

+------------------------------------------------+
|i5 : M = ideal"ab"                              |
|                                                |
|o5 = ideal(a*b)                                 |
|                                                |
|o5 : Ideal of R                                 |
+------------------------------------------------+
|i6 : L1 = standardMonomials M                   |
|                                                |
|        2   2             2                  2  |
|o6 = {{a , b , a*c, b*c, c , a*d, b*d, c*d, d }}|
|                                                |
|o6 : List                                       |
+------------------------------------------------+
|i7 : findWeightVector(M,L1)                     |
+------------------------------------------------+

This command is used in the groebnerFamily, routine.

See also
========

  * groebnerStratum -- compute the ideal where a given
    is a Groebner basis
  * groebnerFamily -- computes families of ideals with a
    specified initial ideal

Ways to use findWeightVector:
=============================

  * "findWeightVector(Ideal,List)"

For the programmer
==================

The object findWeightVector, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:619:0.

groebnerFamily -- computes families of ideals with a specified initial ideal
****************************************************************************

  * Usage:
        J = groebnerFamily M
        J = groebnerFamily(M, L)
  * Inputs:
      * M, an ideal (Macaulay2Doc), which is generated by
        monomials
      * L, a list (Macaulay2Doc), a list of lists of standard
        monomials or smaller standard monomials for the generators of M
  * Optional inputs (Macaulay2Doc),:
      * AllStandard => a Boolean value (Macaulay2Doc), default
        value false
      * Weights => a list (Macaulay2Doc), default value null
      * Variable => a symbol (Macaulay2Doc), default value t, or
        a string (Macaulay2Doc),
  * Outputs:
      * F, an ideal (Macaulay2Doc), the groebner family, an ideal
        in the polynomial ring over the original variables and the parameters

Description
===========

Given a monomial ideal $M$ in a polynomial ring $R$, this computes the
parameter families of homogeneous ideals where $M$ could be their initial
ideal.  These families are obtained from either the standard monomials to the
generators of $M$, or the standard monomials smaller than the generators of $M$
but of the same degree as these generators.  In the former case we obtain a
family of all ideals where $M$ could be their initial ideal.  In the latter
case, we obtain such a family with respect to a given term order.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a,b,c,d];                                                  |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, a*b, b^2)                                               |
|                                                                             |
|             2        2                                                      |
|o2 = ideal (a , a*b, b )                                                     |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : F = groebnerFamily M                                                    |
|                                                                             |
|             2                                      2              2         |
|o3 = ideal (a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d , a*b +  |
|                  1       2       4       5       3      6       7           |
|     ------------------------------------------------------------------------|
|                                           2                2   2            |
|     t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d , b  + t  a*c + |
|      8       9       11       12       10      13       14          15      |
|     ------------------------------------------------------------------------|
|                                    2                2                       |
|     t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d )                      |
|      16       18       19       17      20       21                         |
|                                                                             |
|                ZZ                                                           |
|o3 : Ideal of -----[t , t ..t , t  , t  ..t  , t  , t  ..t  , t ..t , t ..t ,|
|              32003  3   6   7   10   13   14   17   20   21   1   2   4   5 |
|-----------------------------------------------------------------------------|
|t ..t , t  ..t  , t  ..t  , t  ..t  ][a..d]                                  |
| 8   9   11   12   15   16   18   19                                         |
+-----------------------------------------------------------------------------+
|i4 : netList F_*                                                             |
|                                                                             |
|     +---------------------------------------------------------------+       |
|     | 2                                      2              2       |       |
|o4 = |a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d        |       |
|     |      1       2       4       5       3      6       7         |       |
|     +---------------------------------------------------------------+       |
|     |                                            2                2 |       |
|     |a*b + t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d  |       |
|     |       8       9       11       12       10      13       14   |       |
|     +---------------------------------------------------------------+       |
|     | 2                                           2                2|       |
|     |b  + t  a*c + t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d |       |
|     |      15       16       18       19       17      20       21  |       |
|     +---------------------------------------------------------------+       |
+-----------------------------------------------------------------------------+
|i5 : U = ring F                                                              |
|                                                                             |
|o5 = U                                                                       |
|                                                                             |
|o5 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i6 : T = coefficientRing U                                                   |
|                                                                             |
|o6 = T                                                                       |
|                                                                             |
|o6 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i7 : gens T                                                                  |
|                                                                             |
|o7 = {t , t , t , t  , t  , t  , t  , t  , t  , t , t , t , t , t , t , t  , |
|       3   6   7   10   13   14   17   20   21   1   2   4   5   8   9   11  |
|     ------------------------------------------------------------------------|
|     t  , t  , t  , t  , t  }                                                |
|      12   15   16   18   19                                                 |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+
|i8 : gens U                                                                  |
|                                                                             |
|o8 = {a, b, c, d}                                                            |
|                                                                             |
|o8 : List                                                                    |
+-----------------------------------------------------------------------------+

Here, $F$ is the family of homogeneous ideals having $M$ as their initial
ideal, under the term order of the ring of $M$.

The optional argument AllStandard, is boolean, taking the
value $true$ to compute the family of all homogeneous ideals with a given
initial ideal and the value $false$ to compute the family with respect to a
given order.  The default value for this argument is false.

If $L$ is not given, then it is computed using standardMonomials,
(if AllStandard is true), or smallerMonomials,
(if AllStandard is false).

+-----------------------------------------------------------------------------+
|i9 : L = standardMonomials M                                                 |
|                                                                             |
|                  2                  2               2                  2    |
|o9 = {{a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d, d },  |
|     ------------------------------------------------------------------------|
|                 2                  2                                        |
|     {a*c, b*c, c , a*d, b*d, c*d, d }}                                      |
|                                                                             |
|o9 : List                                                                    |
+-----------------------------------------------------------------------------+
|i10 : F2 = groebnerFamily (M, L)                                             |
|                                                                             |
|              2                                      2              2        |
|o10 = ideal (a  + t a*c + t b*c + t a*d + t b*d + t c  + t c*d + t d , a*b + |
|                   1       2       4       5       3      6       7          |
|      -----------------------------------------------------------------------|
|                                            2                2   2           |
|      t a*c + t b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d , b  + t  a*c +|
|       8       9       11       12       10      13       14          15     |
|      -----------------------------------------------------------------------|
|                                     2                2                      |
|      t  b*c + t  a*d + t  b*d + t  c  + t  c*d + t  d )                     |
|       16       18       19       17      20       21                        |
|                                                                             |
|                 ZZ                                                          |
|o10 : Ideal of -----[t , t ..t , t  , t  ..t  , t  , t  ..t  , t ..t , t ..t |
|               32003  3   6   7   10   13   14   17   20   21   1   2   4   5|
|-----------------------------------------------------------------------------|
|, t ..t , t  ..t  , t  ..t  , t  ..t  ][a..d]                                |
|   8   9   11   12   15   16   18   19                                       |
+-----------------------------------------------------------------------------+

Note that $F$ and $F_2$ are the same family, in this case.

This function also works if the ring of $M$ is a skew-commutative ring. This
function produces a family of ideals in the exterior algebra. Similarly, see
groebnerStratum, allows one to find the ideal in the commuting
coefficient variables of the family of ideals having $M$ as lead monomial
ideal.

+-----------------------------------------------------------------------------+
|i11 : kk = ZZ/101                                                            |
|                                                                             |
|o11 = kk                                                                     |
|                                                                             |
|o11 : QuotientRing                                                           |
+-----------------------------------------------------------------------------+
|i12 : E = kk[a,b,c,d,e,SkewCommutative => true]                              |
|                                                                             |
|o12 = E                                                                      |
|                                                                             |
|o12 : PolynomialRing, 5 skew commutative variable(s)                         |
+-----------------------------------------------------------------------------+
|i13 : I = ideal(a*d,a*c,a*b,b*d*e,b*c*e,b*c*d)                               |
|                                                                             |
|o13 = ideal (a*d, a*c, a*b, b*d*e, b*c*e, b*c*d)                             |
|                                                                             |
|o13 : Ideal of E                                                             |
+-----------------------------------------------------------------------------+
|i14 : F1 = groebnerFamily I                                                  |
|                                                                             |
|o14 = ideal (a*d + t b*d + t a*e + t c*d + t b*e + t c*e + t d*e, a*c + t b*c|
|                    1       3       2       4       5       6            7   |
|      -----------------------------------------------------------------------|
|      + t b*d + t  a*e + t c*d + t  b*e + t  c*e + t  d*e, a*b + t  b*c +    |
|         8       10       9       11       12       13            14         |
|      -----------------------------------------------------------------------|
|      t  b*d + t  a*e + t  c*d + t  b*e + t  c*e + t  d*e, b*d*e + t  c*d*e, |
|       15       17       16       18       19       20              21       |
|      -----------------------------------------------------------------------|
|      b*c*e + t  c*d*e, b*c*d + t  c*d*e)                                    |
|               22                23                                          |
|                                                                             |
|o14 : Ideal of kk[t  ..t  , t ..t , t  ..t  , t  , t  , t , t , t , t  , t  .|
|                   19   20   5   6   12   13   16   18   2   4   9   11   14 |
|-----------------------------------------------------------------------------|
|.t  , t  , t  , t , t , t ..t , t  , t  ..t  ][a..e]                         |
|  15   17   23   1   3   7   8   10   21   22                                |
+-----------------------------------------------------------------------------+
|i15 : netList F1_*                                                           |
|                                                                             |
|      +------------------------------------------------------------------+   |
|o15 = |a*d + t b*d + t a*e + t c*d + t b*e + t c*e + t d*e               |   |
|      |       1       3       2       4       5       6                  |   |
|      +------------------------------------------------------------------+   |
|      |a*c + t b*c + t b*d + t  a*e + t c*d + t  b*e + t  c*e + t  d*e   |   |
|      |       7       8       10       9       11       12       13      |   |
|      +------------------------------------------------------------------+   |
|      |a*b + t  b*c + t  b*d + t  a*e + t  c*d + t  b*e + t  c*e + t  d*e|   |
|      |       14       15       17       16       18       19       20   |   |
|      +------------------------------------------------------------------+   |
|      |b*d*e + t  c*d*e                                                  |   |
|      |         21                                                       |   |
|      +------------------------------------------------------------------+   |
|      |b*c*e + t  c*d*e                                                  |   |
|      |         22                                                       |   |
|      +------------------------------------------------------------------+   |
|      |b*c*d + t  c*d*e                                                  |   |
|      |         23                                                       |   |
|      +------------------------------------------------------------------+   |
+-----------------------------------------------------------------------------+
|i16 : F2 = groebnerFamily(I, AllStandard => true)                            |
|                                                                             |
|o16 = ideal (a*d + t a*e + t b*c + t b*d + t c*d + t b*e + t c*e + t d*e, a*c|
|                    4       1       2       3       5       6       7        |
|      -----------------------------------------------------------------------|
|      + t  a*e + t b*c + t b*d + t  c*d + t  b*e + t  c*e + t  d*e, a*b +    |
|         11       8       9       10       12       13       14              |
|      -----------------------------------------------------------------------|
|      t  a*e + t  b*c + t  b*d + t  c*d + t  b*e + t  c*e + t  d*e, b*d*e +  |
|       18       15       16       17       19       20       21              |
|      -----------------------------------------------------------------------|
|      t  c*d*e, b*c*e + t  c*d*e, b*c*d + t  c*d*e)                          |
|       22                23                24                                |
|                                                                             |
|o16 : Ideal of kk[t  ..t  , t ..t , t  ..t  , t  , t  , t , t , t  , t  , t  |
|                   20   21   6   7   13   14   17   19   3   5   10   12   15|
|-----------------------------------------------------------------------------|
|..t  , t  , t  , t ..t , t ..t , t , t  , t  ..t  ][a..e]                    |
|   16   18   24   1   2   8   9   4   11   22   23                           |
+-----------------------------------------------------------------------------+
|i17 : netList F2_*                                                           |
|                                                                             |
|      +------------------------------------------------------------------+   |
|o17 = |a*d + t a*e + t b*c + t b*d + t c*d + t b*e + t c*e + t d*e       |   |
|      |       4       1       2       3       5       6       7          |   |
|      +------------------------------------------------------------------+   |
|      |a*c + t  a*e + t b*c + t b*d + t  c*d + t  b*e + t  c*e + t  d*e  |   |
|      |       11       8       9       10       12       13       14     |   |
|      +------------------------------------------------------------------+   |
|      |a*b + t  a*e + t  b*c + t  b*d + t  c*d + t  b*e + t  c*e + t  d*e|   |
|      |       18       15       16       17       19       20       21   |   |
|      +------------------------------------------------------------------+   |
|      |b*d*e + t  c*d*e                                                  |   |
|      |         22                                                       |   |
|      +------------------------------------------------------------------+   |
|      |b*c*e + t  c*d*e                                                  |   |
|      |         23                                                       |   |
|      +------------------------------------------------------------------+   |
|      |b*c*d + t  c*d*e                                                  |   |
|      |         24                                                       |   |
|      +------------------------------------------------------------------+   |
+-----------------------------------------------------------------------------+
|i18 : J2 = trim groebnerStratum F2                                           |
|                                                                             |
|                                                                             |
|o18 = ideal (t   + t  t  + t  t   - t t  t   + t  t   - t t  t  , t  - t   + |
|              14    24 9    10 11    9 11 22    12 23    8 11 23   7    13   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  t  - t  t  + t  t  + t t   + t  t   - t t t   - t t  t   + t t   -  |
|       24 2    24 8    10 4    3 11    12 22    9 4 22    2 11 22    5 23    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t t t   - t t  t  , t  + t  t  - t t  - t t   + t t t   + t t t  , t   |
|       8 4 23    1 11 23   6    24 1    3 4    5 22    2 4 22    1 4 23   21 |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + t  t   - t  t   + t  t   + t  t   - t  t t   + t  t t   + t  t  t   -|
|         10 18    10 24    16 24    17 11    18 9 22    24 9 22    10 11 22  |
|      -----------------------------------------------------------------------|
|                        2                                                    |
|      t  t  t   - t t  t   - t  t   + t  t   - t  t t   - t  t  t   +        |
|       16 11 22    9 11 22    13 23    19 23    18 8 23    15 11 23          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  t  t   - t t  t  t  , t   - t t   + t t   + t  t   - t  t  + t  t   |
|       12 22 23    8 11 22 23   20    3 18    3 24    15 24    17 4    13 22 |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - t  t   + t  t t   - t  t t   + t  t t   - t  t t   + t  t t   -      |
|         19 22    18 2 22    24 2 22    24 8 22    10 4 22    16 4 22        |
|      -----------------------------------------------------------------------|
|                     2         2          2                                  |
|      t t  t   - t  t   + t t t   + t t  t   + t  t t   - t  t t   + t t t   |
|       3 11 22    12 22    9 4 22    2 11 22    18 1 23    24 1 23    3 4 23 |
|      -----------------------------------------------------------------------|
|                                                                2            |
|      + t  t t   - t t t  t   + t t t  t   + t t  t  t   - t t t  , t  t   - |
|         15 4 23    2 4 22 23    8 4 22 23    1 11 22 23    1 4 23   17 18   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  t   + t  t  t   - t  t  t   - t  t  t   + t  t  t   + t  t  t   -   |
|       17 24    10 18 22    16 18 22    10 24 22    16 24 22    17 11 22     |
|      -----------------------------------------------------------------------|
|            2          2           2           2          3                  |
|      t  t t   + t  t t   + t  t  t   - t  t  t   - t t  t   - t t  t   -    |
|       18 9 22    24 9 22    10 11 22    16 11 22    9 11 22    3 18 23      |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  t  t   + t t  t   + t  t  t   - t  t t   + t  t t  t   - t  t t  t  |
|       15 18 23    3 24 23    15 24 23    17 4 23    18 2 22 23    24 2 22 23|
|      -----------------------------------------------------------------------|
|                                                                             |
|      - t  t t  t   + t  t t  t   - t  t t  t   + t  t t  t   - t t  t  t   -|
|         18 8 22 23    24 8 22 23    10 4 22 23    16 4 22 23    3 11 22 23  |
|      -----------------------------------------------------------------------|
|                          2             2             2             2        |
|      t  t  t  t   + t t t  t   + t t  t  t   - t t  t  t   + t  t t   -     |
|       15 11 22 23    9 4 22 23    2 11 22 23    8 11 22 23    18 1 23       |
|      -----------------------------------------------------------------------|
|            2         2          2            2            2             2   |
|      t  t t   + t t t   + t  t t   - t t t  t   + t t t  t   + t t  t  t   -|
|       24 1 23    3 4 23    15 4 23    2 4 22 23    8 4 22 23    1 11 22 23  |
|      -----------------------------------------------------------------------|
|           3                                                                 |
|      t t t  , t  t   - t  t t  - t  t t   + t  t  t   - t  t  t   -         |
|       1 4 23   17 12    17 9 4    17 8 11    10 12 22    12 16 22           |
|      -----------------------------------------------------------------------|
|                                                                  2          |
|      t  t t t   + t  t t t   - t  t t  t   + t  t t  t   - t  t t   +       |
|       10 9 4 22    16 9 4 22    10 8 11 22    16 8 11 22    12 9 22         |
|      -----------------------------------------------------------------------|
|       2   2            2                                                    |
|      t t t   + t t t  t   - t t  t   - t  t  t   + t t t t   + t  t t t   + |
|       9 4 22    8 9 11 22    3 12 23    12 15 23    3 9 4 23    15 9 4 23   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t t t  t   + t  t t  t   + t  t t  t   - t  t t  t   - t t t t  t   +  |
|       3 8 11 23    15 8 11 23    12 2 22 23    12 8 22 23    2 9 4 22 23    |
|      -----------------------------------------------------------------------|
|                                      2                  2           2       |
|      t t t t  t   - t t t  t  t   + t t  t  t   + t  t t   - t t t t   -    |
|       8 9 4 22 23    2 8 11 22 23    8 11 22 23    12 1 23    1 9 4 23      |
|      -----------------------------------------------------------------------|
|              2                                                              |
|      t t t  t  , t  t  - t  t t  - t  t t   + t t  t   - t t  t   -         |
|       1 8 11 23   17 5    17 2 4    17 1 11    5 10 22    5 16 22           |
|      -----------------------------------------------------------------------|
|                                                                 2           |
|      t  t t t   + t  t t t   - t  t t  t   + t  t t  t   - t t t   +        |
|       10 2 4 22    16 2 4 22    10 1 11 22    16 1 11 22    5 9 22          |
|      -----------------------------------------------------------------------|
|             2            2                                                  |
|      t t t t   + t t t  t   - t t t   - t t  t   + t t t t   + t  t t t   + |
|       2 9 4 22    1 9 11 22    3 5 23    5 15 23    3 2 4 23    15 2 4 23   |
|      -----------------------------------------------------------------------|
|                                                            2                |
|      t t t  t   + t  t t  t   + t t t  t   - t t t  t   - t t t  t   +      |
|       3 1 11 23    15 1 11 23    5 2 22 23    5 8 22 23    2 4 22 23        |
|      -----------------------------------------------------------------------|
|                                                          2           2      |
|      t t t t  t   - t t t  t  t   + t t t  t  t   + t t t   - t t t t   -   |
|       2 8 4 22 23    1 2 11 22 23    1 8 11 22 23    5 1 23    1 2 4 23     |
|      -----------------------------------------------------------------------|
|       2    2                                                                |
|      t t  t  , t  t   - t  t   + t  t  t  - t  t  t  + t  t  t  + t  t  t   |
|       1 11 23   13 17    17 19    17 24 8    17 10 4    17 16 4    17 15 11 |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + t  t  t   - t  t  t   - t  t  t   + t  t  t   + t  t  t t   -        |
|         13 10 22    19 10 22    13 16 22    19 16 22    10 24 8 22          |
|      -----------------------------------------------------------------------|
|                     2                         2                             |
|      t  t  t t   - t  t t   + 2t  t  t t   - t  t t   + t  t  t  t   -      |
|       16 24 8 22    10 4 22     10 16 4 22    16 4 22    10 15 11 22        |
|      -----------------------------------------------------------------------|
|                                         2          2            2           |
|      t  t  t  t   - t  t t  t   - t  t t   + t  t t   - t  t t t   +        |
|       15 16 11 22    17 8 11 22    13 9 22    19 9 22    24 8 9 22          |
|      -----------------------------------------------------------------------|
|              2            2             2             2             2       |
|      t  t t t   - t  t t t   - t  t t  t   + t  t t  t   - t  t t  t   +    |
|       10 9 4 22    16 9 4 22    10 8 11 22    16 8 11 22    15 9 11 22      |
|      -----------------------------------------------------------------------|
|              3                                                              |
|      t t t  t   - t  t t   + t  t t   - t  t  t   + t  t  t   - t t  t t   -|
|       8 9 11 22    13 3 23    19 3 23    13 15 23    19 15 23    3 24 8 23  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  t  t t   + t t  t t   + t  t  t t   - t t  t t   - t  t  t t   +    |
|       15 24 8 23    3 10 4 23    10 15 4 23    3 16 4 23    15 16 4 23      |
|      -----------------------------------------------------------------------|
|                                  2                                          |
|      t  t t t   - t t  t  t   - t  t  t   + t  t t  t   - t  t t  t   -     |
|       17 8 4 23    3 15 11 23    15 11 23    13 2 22 23    19 2 22 23       |
|      -----------------------------------------------------------------------|
|                                                      2                      |
|      t  t t  t   + t  t t  t   + t  t t t  t   - t  t t  t   - t  t t t  t  |
|       13 8 22 23    19 8 22 23    24 2 8 22 23    24 8 22 23    10 2 4 22 23|
|      -----------------------------------------------------------------------|
|                                                                             |
|      + t  t t t  t   + 2t  t t t  t   - 2t  t t t  t   + t  t t  t  t   +   |
|         16 2 4 22 23     10 8 4 22 23     16 8 4 22 23    15 2 11 22 23     |
|      -----------------------------------------------------------------------|
|                             2               2        2    2             2   |
|      t t t  t  t   - t t t t  t   - t t t  t  t   + t t  t  t   + t  t t   -|
|       3 8 11 22 23    8 9 4 22 23    2 8 11 22 23    8 11 22 23    13 1 23  |
|      -----------------------------------------------------------------------|
|            2            2            2            2           2             |
|      t  t t   + t  t t t   - t  t t t   + t  t t t   - t t t t   -          |
|       19 1 23    24 1 8 23    10 1 4 23    16 1 4 23    3 8 4 23            |
|      -----------------------------------------------------------------------|
|              2             2              2     2      2               2    |
|      t  t t t   + t  t t  t   + t t t t  t   - t t t  t   - t t t  t  t   + |
|       15 8 4 23    15 1 11 23    2 8 4 22 23    8 4 22 23    1 8 11 22 23   |
|      -----------------------------------------------------------------------|
|             3                                                               |
|      t t t t  )                                                             |
|       1 8 4 23                                                              |
|                                                                             |
|o18 : Ideal of kk[t  ..t  , t ..t , t  ..t  , t  , t  , t , t , t  , t  , t  |
|                   20   21   6   7   13   14   17   19   3   5   10   12   15|
|-----------------------------------------------------------------------------|
|..t  , t  , t  , t ..t , t ..t , t , t  , t  ..t  ]                          |
|   16   18   24   1   2   8   9   4   11   22   23                           |
+-----------------------------------------------------------------------------+
|i19 : C2 = decompose J2                                                      |
|                                                                             |
|                                         2                                   |
|o19 = {ideal (t   + t  t   - t  t   - t t   - t t   - t  t   + t t  t   -    |
|               17    10 22    16 22    9 22    3 23    15 23    2 22 23      |
|      -----------------------------------------------------------------------|
|                    2                                                        |
|      t t  t   + t t  , t   + t  t  + t  t   - t t  t   + t  t   - t t  t  , |
|       8 22 23    1 23   14    24 9    10 11    9 11 22    12 23    8 11 23  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t  - t   + t  t  - t  t  + t  t  + t t   + t  t   - t t t   - t t  t   |
|       7    13    24 2    24 8    10 4    3 11    12 22    9 4 22    2 11 22 |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + t t   - t t t   - t t  t  , t  + t  t  - t t  - t t   + t t t   +    |
|         5 23    8 4 23    1 11 23   6    24 1    3 4    5 22    2 4 22      |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t t t  , t   + t  t   - t  t   + t  t   - t  t t   + t  t t   - t  t   |
|       1 4 23   21    10 18    10 24    16 24    18 9 22    24 9 22    13 23 |
|      -----------------------------------------------------------------------|
|                                                                       2     |
|      + t  t   - t  t t   + t t  t   + t  t  t   - t t  t  t   - t t  t  ,   |
|         19 23    18 8 23    3 11 23    12 22 23    2 11 22 23    1 11 23    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      t   - t t   + t t   + t  t   + t  t   - t  t   + t  t t   - t  t t   + |
|       20    3 18    3 24    15 24    13 22    19 22    18 2 22    24 2 22   |
|      -----------------------------------------------------------------------|
|                                2          2                                 |
|      t  t t   - t t  t   - t  t   + t t  t   + t  t t   - t  t t   +        |
|       24 8 22    3 11 22    12 22    2 11 22    18 1 23    24 1 23          |
|      -----------------------------------------------------------------------|
|      t t  t  t  ), ideal (t   - t   + t  t   - t t  , t   - t t  - t t  , t |
|       1 11 22 23           18    24    11 22    4 23   12    9 4    8 11   5|
|      -----------------------------------------------------------------------|
|      - t t  - t t  , t   + t  t  + t  t   - t t  t   + t t t  , t   - t   + |
|         2 4    1 11   14    24 9    10 11    9 11 22    9 4 23   13    19   |
|      -----------------------------------------------------------------------|
|      t  t  - t  t  + t  t  + t  t   - t t  t   + t t t  , t  - t   + t  t  +|
|       24 8    10 4    16 4    15 11    8 11 22    8 4 23   7    19    24 2  |
|      -----------------------------------------------------------------------|
|      t  t  + t t   + t  t   - t t  t   + t t t  , t  + t  t  - t t  -       |
|       16 4    3 11    15 11    2 11 22    2 4 23   6    24 1    3 4         |
|      -----------------------------------------------------------------------|
|      t t  t   + t t t  , t   + t  t   + t  t   - t  t  t   + t  t t  , t   +|
|       1 11 22    1 4 23   21    16 24    17 11    16 11 22    16 4 23   20  |
|      -----------------------------------------------------------------------|
|      t  t   - t  t  - t  t  t   + t  t t  )}                                |
|       15 24    17 4    15 11 22    15 4 23                                  |
|                                                                             |
|o19 : List                                                                   |
+-----------------------------------------------------------------------------+
|i20 : netList C2_0_*                                                         |
|                                                                             |
|      +----------------------------------------------------------------------|
|      |                           2                                          |
|o20 = |t   + t  t   - t  t   - t t   - t t   - t  t   + t t  t   - t t  t   +|
|      | 17    10 22    16 22    9 22    3 23    15 23    2 22 23    8 22 23  |
|      +----------------------------------------------------------------------|
|      |t   + t  t  + t  t   - t t  t   + t  t   - t t  t                     |
|      | 14    24 9    10 11    9 11 22    12 23    8 11 23                   |
|      +----------------------------------------------------------------------|
|      |t  - t   + t  t  - t  t  + t  t  + t t   + t  t   - t t t   - t t  t  |
|      | 7    13    24 2    24 8    10 4    3 11    12 22    9 4 22    2 11 22|
|      +----------------------------------------------------------------------|
|      |t  + t  t  - t t  - t t   + t t t   + t t t                           |
|      | 6    24 1    3 4    5 22    2 4 22    1 4 23                         |
|      +----------------------------------------------------------------------|
|      |                                                                      |
|      |t   + t  t   - t  t   + t  t   - t  t t   + t  t t   - t  t   + t  t  |
|      | 21    10 18    10 24    16 24    18 9 22    24 9 22    13 23    19 23|
|      +----------------------------------------------------------------------|
|      |                                                                      |
|      |t   - t t   + t t   + t  t   + t  t   - t  t   + t  t t   - t  t t   +|
|      | 20    3 18    3 24    15 24    13 22    19 22    18 2 22    24 2 22  |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|----------------------------------------------------------------------------+|
|    2                                                                       ||
| t t                                                                        ||
|  1 23                                                                      ||
|----------------------------------------------------------------------------+|
|                                                                            ||
|                                                                            ||
|----------------------------------------------------------------------------+|
| + t t   - t t t   - t t  t                                                 ||
|    5 23    8 4 23    1 11 23                                               ||
|----------------------------------------------------------------------------+|
|                                                                            ||
|                                                                            ||
|----------------------------------------------------------------------------+|
|                                                         2                  ||
| - t  t t   + t t  t   + t  t  t   - t t  t  t   - t t  t                   ||
|    18 8 23    3 11 23    12 22 23    2 11 22 23    1 11 23                 ||
|----------------------------------------------------------------------------+|
|                           2          2                                     ||
| t  t t   - t t  t   - t  t   + t t  t   + t  t t   - t  t t   + t t  t  t  ||
|  24 8 22    3 11 22    12 22    2 11 22    18 1 23    24 1 23    1 11 22 23||
|----------------------------------------------------------------------------+|
+-----------------------------------------------------------------------------+
|i21 : netList C2_1_*                                                         |
|                                                                             |
|      +---------------------------------------------------------------+      |
|o21 = |t   - t   + t  t   - t t                                       |      |
|      | 18    24    11 22    4 23                                     |      |
|      +---------------------------------------------------------------+      |
|      |t   - t t  - t t                                               |      |
|      | 12    9 4    8 11                                             |      |
|      +---------------------------------------------------------------+      |
|      |t  - t t  - t t                                                |      |
|      | 5    2 4    1 11                                              |      |
|      +---------------------------------------------------------------+      |
|      |t   + t  t  + t  t   - t t  t   + t t t                        |      |
|      | 14    24 9    10 11    9 11 22    9 4 23                      |      |
|      +---------------------------------------------------------------+      |
|      |t   - t   + t  t  - t  t  + t  t  + t  t   - t t  t   + t t t  |      |
|      | 13    19    24 8    10 4    16 4    15 11    8 11 22    8 4 23|      |
|      +---------------------------------------------------------------+      |
|      |t  - t   + t  t  + t  t  + t t   + t  t   - t t  t   + t t t   |      |
|      | 7    19    24 2    16 4    3 11    15 11    2 11 22    2 4 23 |      |
|      +---------------------------------------------------------------+      |
|      |t  + t  t  - t t  - t t  t   + t t t                           |      |
|      | 6    24 1    3 4    1 11 22    1 4 23                         |      |
|      +---------------------------------------------------------------+      |
|      |t   + t  t   + t  t   - t  t  t   + t  t t                     |      |
|      | 21    16 24    17 11    16 11 22    16 4 23                   |      |
|      +---------------------------------------------------------------+      |
|      |t   + t  t   - t  t  - t  t  t   + t  t t                      |      |
|      | 20    15 24    17 4    15 11 22    15 4 23                    |      |
|      +---------------------------------------------------------------+      |
+-----------------------------------------------------------------------------+

See also
========

  * groebnerStratum -- compute the ideal where a given
    is a Groebner basis
  * smallerMonomials -- returns the standard monomials
    smaller but of the same degree as given monomial(s)
  * standardMonomials -- computes standard monomials

Ways to use groebnerFamily:
===========================

  * "groebnerFamily(Ideal)"
  * "groebnerFamily(Ideal,List)"

For the programmer
==================

The object groebnerFamily, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:758:0.

groebnerStratum -- compute the ideal where a given is a Groebner basis
**********************************************************************

  * Usage:
        J = groebnerStratum F
  * Inputs:
      * F, an ideal (Macaulay2Doc), An ideal constructed from
        groebnerFamily,
  * Optional inputs (Macaulay2Doc),:
      * Minimalize => ..., default value false
  * Outputs:
      * J, an ideal (Macaulay2Doc), the ideal of the Groebner
        stratum

Description
===========

Given a monomial ideal $M$, this command returns a family of ideals having $M$
as an initial ideal, and conditions on the parameters so that the family is
flat.  If the optional input AllStandard, is specified as
$true$, the family of all homogeneous ideals having $M$ as an initial ideal is
computed, and if it is specified as $false$ the family of all homogeneous
ideals having $M$ as an initial ideal with respect to the given term order is
computed.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a,b,c]                                                     |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, a*b, b^2)                                               |
|                                                                             |
|             2        2                                                      |
|o2 = ideal (a , a*b, b )                                                     |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : F = groebnerFamily M                                                    |
|                                                                             |
|             2                      2                           2   2        |
|o3 = ideal (a  + t a*c + t b*c + t c , a*b + t a*c + t b*c + t c , b  + t a*c|
|                  1       2       3           4       5       6          7   |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     + t b*c + t c )                                                         |
|        8       9                                                            |
|                                                                             |
|                ZZ                                                           |
|o3 : Ideal of -----[t , t , t , t ..t , t ..t , t ..t ][a..c]                |
|              32003  3   6   9   1   2   4   5   7   8                       |
+-----------------------------------------------------------------------------+
|i4 : J = trim groebnerStratum F                                              |
|                                                                             |
|                  2                                                          |
|o4 = ideal (t  + t  - t t  + t t  - t t , t  - t t  + t t , t  + t t  - t t  |
|             9    4    1 7    5 7    4 8   6    4 5    2 7   3    2 4    1 5 |
|     ------------------------------------------------------------------------|
|        2                                                                    |
|     + t  - t t )                                                            |
|        5    2 8                                                             |
|                                                                             |
|                ZZ                                                           |
|o4 : Ideal of -----[t , t , t , t ..t , t ..t , t ..t ]                      |
|              32003  3   6   9   1   2   4   5   7   8                       |
+-----------------------------------------------------------------------------+

In this example, F is the universal family, and L is the ideal giving the
conditions on the parameters.  In general, several of the parameters are
unnecessary. Note that H is an ideal in a ring with far fewer parameters. This
is because a maximal set of eliminable parameters from the original ideal of
conditions on parameters have been eliminated.  If the full ideal in the
polynomial ring over all the parameters is desired, set the optional argument
$Minimalize$ to false.

+-----------------------------------------------------------------------------+
|i5 : J2 = trim groebnerStratum(F, Minimalize => false)                       |
|                                                                             |
|                  2                                                          |
|o5 = ideal (t  + t  - t t  + t t  - t t , t  - t t  + t t , t  + t t  - t t  |
|             9    4    1 7    5 7    4 8   6    4 5    2 7   3    2 4    1 5 |
|     ------------------------------------------------------------------------|
|        2                                                                    |
|     + t  - t t )                                                            |
|        5    2 8                                                             |
|                                                                             |
|                ZZ                                                           |
|o5 : Ideal of -----[t , t , t , t ..t , t ..t , t ..t ]                      |
|              32003  3   6   9   1   2   4   5   7   8                       |
+-----------------------------------------------------------------------------+
|i6 : netList J_*                                                             |
|                                                                             |
|     +----------------------------+                                          |
|     |      2                     |                                          |
|o6 = |t  + t  - t t  + t t  - t t |                                          |
|     | 9    4    1 7    5 7    4 8|                                          |
|     +----------------------------+                                          |
|     |t  - t t  + t t             |                                          |
|     | 6    4 5    2 7            |                                          |
|     +----------------------------+                                          |
|     |                    2       |                                          |
|     |t  + t t  - t t  + t  - t t |                                          |
|     | 3    2 4    1 5    5    2 8|                                          |
|     +----------------------------+                                          |
+-----------------------------------------------------------------------------+
|i7 : netList J2_*                                                            |
|                                                                             |
|     +----------------------------+                                          |
|     |      2                     |                                          |
|o7 = |t  + t  - t t  + t t  - t t |                                          |
|     | 9    4    1 7    5 7    4 8|                                          |
|     +----------------------------+                                          |
|     |t  - t t  + t t             |                                          |
|     | 6    4 5    2 7            |                                          |
|     +----------------------------+                                          |
|     |                    2       |                                          |
|     |t  + t t  - t t  + t  - t t |                                          |
|     | 3    2 4    1 5    5    2 8|                                          |
|     +----------------------------+                                          |
+-----------------------------------------------------------------------------+

Notice that the parameters $t_3$, $t_6$, and $t_9$ are clearly eliminable.

See also
========

  * groebnerFamily -- computes families of ideals with a
    specified initial ideal

Ways to use groebnerStratum:
============================

  * "groebnerStratum(Ideal)"

For the programmer
==================

The object groebnerStratum, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:805:0.

linearPart -- returns linear part of a polynomial
*************************************************

  * Usage:
        l = linearPart(f)
  * Inputs:
      * f, a ring element (Macaulay2Doc), f is a polynomial
  * Outputs:
      * l, a ring element (Macaulay2Doc), l is the sum of
        the terms of f whose monomial is a variable

Description
===========

+----------------------------------------+
|i1 : R = ZZ/32003[a,b,c];               |
+----------------------------------------+
|i2 : f = a^2 + 3*b + 5*a*b*c + 2*a +b^2;|
+----------------------------------------+
|i3 : linearPart f                       |
|                                        |
|o3 = 2a + 3b                            |
|                                        |
|o3 : R                                  |
+----------------------------------------+

If we have a polynomial ring over another polynomial ring, the command only
looks at the exponents on the new set of variables.

+--------------------------------+
|i4 : S = ZZ/32003[a,b,c][x,y,z];|
+--------------------------------+
|i5 : g = a*b*x + 3*y + a + b^2  |
|                                |
|                   2            |
|o5 = a*b*x + 3y + b  + a        |
|                                |
|o5 : S                          |
+--------------------------------+
|i6 : linearPart g               |
|                                |
|o6 = a*b*x + 3y                 |
|                                |
|o6 : S                          |
+--------------------------------+

Note that in the previous example, linearPart g does not include a but does
include a*b*x

For the programmer
==================

The object linearPart, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:834:0.

Minimalize -- boolean option for determining whether excess parameters will be eliminated
*****************************************************************************************

Description
===========

This an optional input for the groebnerStratum,
function.  It takes values $true$ and $false$.  When assigned the value $true$,
eliminable parameters are eliminated to obtain the groebner basin / stratum in
a smaller ring.  When assigned the value $false$, no parameters are eliminated,
and the groebner basin / stratum is computed in a ring containing all the
parameters.

See also
========

  * groebnerStratum -- compute the ideal where a given
    is a Groebner basis

Functions with optional argument named Minimalize:
==================================================

  * "groebnerStratum(...,Minimalize=>...)" -- see groebnerStratum,
    -- compute the ideal where a given is a Groebner basis

For the programmer
==================

The object Minimalize, is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:1059:0.

nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal
**********************************************************************************

  * Usage:
        (C, H) = nonminimalMaps I
  * Inputs:
      * I, an ideal (Macaulay2Doc), in a polynomial ring $S$ over
        a base field or coefficient ring $A$.  The lead terms of the generators
        of $I$ should be the initial ideal of $I$, and should be monic.
  * Outputs:
      * C, a chain complex (OldChainComplexes), A complex
        over a polynomial ring where any parameters in the base ring are set to
        have degree 0, and the variables of the ring of $I$ are set to have
        degree one.
      * H, a hash table (Macaulay2Doc), Whose keys describe
        which submatrix in the resolution this is, and whose values are those
        submatrices (placed into the original coefficient ring $A$)

Description
===========

The Schreyer resolution of $I$ (which is generally non-minimal) is computed.
The nonminimal parts are the submatrices in this resolution which do not
involve the variables in $S$.  They are elements in the base ring $A$.  For
instance, H#(\ell, d) is the submatrix of the matrix from $C_{\ell+1} \to
C_{\ell}$ sending degree $d$ to degree $d$.

The ranks of these matrices for a specific parameter value determine exactly
the minimal Betti table for the ideal $I$, evaluated at that parameter point.

Now for our example.

+-----------------------------------------------------------------------------+
|i1 : kk = ZZ/101;                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = kk[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i3 : F = groebnerFamily ideal"a2,ab,ac,b2,bc2,c3"                            |
|                                                                             |
|             2                      2                      2                 |
|o3 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +  |
|                  1       3       2      4       5       6           7       |
|     ------------------------------------------------------------------------|
|                2                         2                              2   |
|     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  + |
|      9       8      10       11       12           13       15       14     |
|     ------------------------------------------------------------------------|
|                           2   2                         2                   |
|     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d |
|      16       17       18          19       21       20      22       23    |
|     ------------------------------------------------------------------------|
|           2     2                    2       2          2         2       3 |
|     + t  d , b*c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d ,|
|        24            25         27        26       28        29        30   |
|     ------------------------------------------------------------------------|
|      3                    2       2          2         2       3            |
|     c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d )           |
|           31         33        32       34        35        36              |
|                                                                             |
|o3 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , |
|                  6   12   5   30   18   4   24   36   11   2   29   3   10  |
|-----------------------------------------------------------------------------|
|t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,  |
| 17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   |
|-----------------------------------------------------------------------------|
|t  , t  , t  , t  , t  , t  , t  ][a..d]                                     |
| 25   32   13   21   33   19   31                                            |
+-----------------------------------------------------------------------------+
|i4 : (C, H) = nonminimalMaps F;                                              |
+-----------------------------------------------------------------------------+
|i5 : betti(C, Weights => {1,1,1,1})                                          |
|                                                                             |
|            0 1  2 3 4                                                       |
|o5 = total: 1 6 10 6 1                                                       |
|         0: 1 .  . . .                                                       |
|         1: . 4  4 2 .                                                       |
|         2: . 2  5 3 1                                                       |
|         3: . .  1 1 .                                                       |
|                                                                             |
|o5 : BettiTally                                                              |
+-----------------------------------------------------------------------------+

We see that there are 4 maps that are nonminimal (of sizes $2 \times 4$, $5
\times 2$, $1 \times 3$, and $1 \times 1$).

+-----------------------------------------------------------------------------+
|i6 : keys H                                                                  |
|                                                                             |
|o6 = {(3, 4), (3, 5), (4, 6), (2, 3)}                                        |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+
|i7 : H#(2,3)                                                                 |
|                                                                             |
|o7 = {3} | -t_8-t_20t_13      t_7t_20-t_14t_20+t_20t_13t_19                  |
|     {3} | -t_7+t_14-t_13t_19 -t_8-t_20t_13+t_7t_19-t_14t_19+t_13t_19^2      |
|     ------------------------------------------------------------------------|
|     -t_2-t_14^2+t_20t_13^2    -t_8t_14+t_1t_20+t_7t_20t_13             |    |
|     -t_1-2t_14t_13+t_13^2t_19 -t_2-t_7t_14-t_8t_13+t_1t_19+t_7t_13t_19 |    |
|                                                                             |
|                                                                             |
|o7 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t|
|                 6   12   5   30   18   4   24   36   11   2   29   3   10   |
|-----------------------------------------------------------------------------|
|                                                                             |
|  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,   |
|17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20    |
|-----------------------------------------------------------------------------|
|                                   2                                         |
|t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , |
| 25   32   13   21   33   19   31             6   12   5   30   18   4   24  |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  ,   |
| 36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26    |
|-----------------------------------------------------------------------------|
|                                                                4            |
|t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])             |
| 34   7   14   27   15   20   25   32   13   21   33   19   31               |
+-----------------------------------------------------------------------------+
|i8 : H#(3,4)                                                                 |
|                                                                             |
|o8 = {4} | -t_20                                                             |
|     {4} | -1                                                                |
|     {4} | t_8+t_20t_13-t_7t_19+t_14t_19-t_13t_19^2                          |
|     {4} | -t_7+t_14-t_13t_19                                                |
|     {4} | 0                                                                 |
|     ------------------------------------------------------------------------|
|     -t_8                                    |                               |
|     t_13                                    |                               |
|     t_2+t_7t_14+t_8t_13-t_1t_19-t_7t_13t_19 |                               |
|     -t_1-2t_14t_13+t_13^2t_19               |                               |
|     t_7-t_14+t_13t_19                       |                               |
|                                                                             |
|                                                                             |
|o8 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t|
|                 6   12   5   30   18   4   24   36   11   2   29   3   10   |
|-----------------------------------------------------------------------------|
|                                                                             |
|  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,   |
|17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20    |
|-----------------------------------------------------------------------------|
|                                   5                                         |
|t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , |
| 25   32   13   21   33   19   31             6   12   5   30   18   4   24  |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  ,   |
| 36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26    |
|-----------------------------------------------------------------------------|
|                                                                2            |
|t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])             |
| 34   7   14   27   15   20   25   32   13   21   33   19   31               |
+-----------------------------------------------------------------------------+
|i9 : H#(3,5)                                                                 |
|                                                                             |
|o9 = {5} | -1 t_13 -t_14 |                                                   |
|                                                                             |
|                                                                             |
|o9 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  ,  |
|                 6   12   5   30   18   4   24   36   11   2   29   3   10   |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,  |
| 17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   |
|-----------------------------------------------------------------------------|
|                                   1                                         |
|t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , |
| 25   32   13   21   33   19   31             6   12   5   30   18   4   24  |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  ,   |
| 36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26    |
|-----------------------------------------------------------------------------|
|                                                                3            |
|t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])             |
| 34   7   14   27   15   20   25   32   13   21   33   19   31               |
+-----------------------------------------------------------------------------+
|i10 : H#(4,6)                                                                |
|                                                                             |
|o10 = {6} | -1 |                                                             |
|                                                                             |
|                                                                             |
|o10 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , |
|                  6   12   5   30   18   4   24   36   11   2   29   3   10  |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,  |
| 17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   |
|-----------------------------------------------------------------------------|
|                                   1                                         |
|t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , |
| 25   32   13   21   33   19   31             6   12   5   30   18   4   24  |
|-----------------------------------------------------------------------------|
|                                                                             |
|t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  ,   |
| 36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26    |
|-----------------------------------------------------------------------------|
|                                                                1            |
|t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])             |
| 34   7   14   27   15   20   25   32   13   21   33   19   31               |
+-----------------------------------------------------------------------------+

Let's impose the condition that the map H#(2,3) vanishes (so has rank 0). The
Betti diagram of such ideals is not the one for a set of 6 generic points in
$\PP^3$.

+-----------------------------------------------------------------------------+
|i11 : J = trim(minors(1, H#(2,3)) + groebnerStratum F);                      |
|                                                                             |
|o11 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  ,|
|                   6   12   5   30   18   4   24   36   11   2   29   3   10 |
|-----------------------------------------------------------------------------|
|t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  ,  |
| 17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   |
|-----------------------------------------------------------------------------|
|t  , t  , t  , t  , t  , t  , t  ]                                           |
| 25   32   13   21   33   19   31                                            |
+-----------------------------------------------------------------------------+
|i12 : compsJ = decompose J;                                                  |
+-----------------------------------------------------------------------------+
|i13 : #compsJ                                                                |
|                                                                             |
|o13 = 2                                                                      |
+-----------------------------------------------------------------------------+
|i14 : pt1 = randomPointOnRationalVariety compsJ_0                            |
|                                                                             |
|o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 |
|      -----------------------------------------------------------------------|
|      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |             |
|                                                                             |
|               1       36                                                    |
|o14 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i15 : pt2 = randomPointOnRationalVariety compsJ_1                            |
|                                                                             |
|o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47  |
|      -----------------------------------------------------------------------|
|      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |                 |
|                                                                             |
|               1       36                                                    |
|o15 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i16 : F1 = sub(F, (vars S)|pt1)                                              |
|                                                                             |
|              2              2                             2                 |
|o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +  |
|      -----------------------------------------------------------------------|
|         2                              2                  2                 |
|      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d |
|      -----------------------------------------------------------------------|
|                  2   2              2                              2     2  |
|      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -|
|      -----------------------------------------------------------------------|
|                   2         2        2        2      3   3                2 |
|      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d|
|      -----------------------------------------------------------------------|
|             2        2        2      3                                      |
|      + 19a*d  - 38b*d  - 29c*d  - 26d )                                     |
|                                                                             |
|o16 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i17 : betti res F1                                                           |
|                                                                             |
|             0 1 2 3                                                         |
|o17 = total: 1 6 8 3                                                         |
|          0: 1 . . .                                                         |
|          1: . 4 4 1                                                         |
|          2: . 2 4 2                                                         |
|                                                                             |
|o17 : BettiTally                                                             |
+-----------------------------------------------------------------------------+
|i18 : F2 = sub(F, (vars S)|pt2)                                              |
|                                                                             |
|              2             2                              2                 |
|o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +  |
|      -----------------------------------------------------------------------|
|        2                              2                   2                 |
|      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d |
|      -----------------------------------------------------------------------|
|                 2   2              2                      2     2           |
|      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d |
|      -----------------------------------------------------------------------|
|           2         2        2       2      3   3               2         2 |
|      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d  |
|      -----------------------------------------------------------------------|
|             2        2    3                                                 |
|      + 38b*d  - 39c*d  - d )                                                |
|                                                                             |
|o18 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i19 : betti res F2                                                           |
|                                                                             |
|             0 1 2 3                                                         |
|o19 = total: 1 6 8 3                                                         |
|          0: 1 . . .                                                         |
|          1: . 4 4 1                                                         |
|          2: . 2 4 2                                                         |
|                                                                             |
|o19 : BettiTally                                                             |
+-----------------------------------------------------------------------------+

What are the ideals F1 and F2?

+-----------------------------------------------------------------------------+
|i20 : netList decompose F1                                                   |
|                                                                             |
|      +----------------------------------------------------------------------|
|o20 = |ideal (c + 39d, b + 27d, a - 18d)                                     |
|      +----------------------------------------------------------------------|
|      |                            2              2                     2   3|
|      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|-----------------------------------------------------------------------------|
|                2        2        2      3     2                2         2  |
| - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  +|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|------+                                                                      |
|      |                                                                      |
|------+                                                                      |
|    3 |                                                                      |
| 14d )|                                                                      |
|------+                                                                      |
+-----------------------------------------------------------------------------+
|i21 : netList decompose F2                                                   |
|                                                                             |
|      +-------------------------------------------------------+              |
|o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |              |
|      +-------------------------------------------------------+              |
|      |ideal (c + 43d, b - 47d, a - 27d)                      |              |
|      +-------------------------------------------------------+              |
|      |ideal (c + 24d, b - 49d, a)                            |              |
|      +-------------------------------------------------------+              |
|      |ideal (c + 14d, b + 31d, a - 16d)                      |              |
|      +-------------------------------------------------------+              |
|      |                                      2              2 |              |
|      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|              |
|      +-------------------------------------------------------+              |
+-----------------------------------------------------------------------------+

We can determine what these represent.  One should be a set of 6 points, where
5 lie on a plane.  The other should be 6 points with 3 points on one line, and
the other 3 points on a skew line.

See also
========

  * randomPointOnRationalVariety
    -- find a random point on a
    variety that can be detected to be rational

Ways to use nonminimalMaps:
===========================

  * "nonminimalMaps(Ideal)"

For the programmer
==================

The object nonminimalMaps, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:1012:0.

randomPointOnRationalVariety(Ideal) -- find a random point on a variety that can be detected to be rational
***********************************************************************************************************

  * Function: randomPointOnRationalVariety,
  * Usage:
        randomPointOnRationalVariety I
        randomPointOnRationalVariety
  * Inputs:
      * I, an ideal (Macaulay2Doc), An ideal in a polynomial ring
        $S$ over a field, which defines a prime ideal
  * Outputs:
      * a matrix (Macaulay2Doc), A one row matrix over the base
        field of $S$, representing a randomly chosen point on the zero locus of
        $I$.  null is returned in the case when the routine cannot determine if
        the variety is rational and irreducible.

Description
===========

As a first example, we find a random point on the Veronese surface in $\PP^5$.

+-----------------------------------------------------------------------------+
|i1 : kk = ZZ/101;                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = kk[a..f];                                                           |
+-----------------------------------------------------------------------------+
|i3 : I = minors(2, genericSymmetricMatrix(S, 3))                             |
|                                                                             |
|               2                                                  2          |
|o3 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - |
|     ------------------------------------------------------------------------|
|                                             2                               |
|     c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)                        |
|                                                                             |
|o3 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i4 : pt = randomPointOnRationalVariety I                                     |
|                                                                             |
|o4 = | 1 49 24 -23 -36 -30 |                                                 |
|                                                                             |
|              1       6                                                      |
|o4 : Matrix kk  <-- kk                                                       |
+-----------------------------------------------------------------------------+
|i5 : sub(I, pt) == 0                                                         |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i6 : S = kk[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i7 : F = groebnerFamily ideal"a2,ab,ac,b2"                                   |
|                                                                             |
|             2                      2                      2                 |
|o7 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +  |
|                  1       3       2      4       5       6           7       |
|     ------------------------------------------------------------------------|
|                2                         2                              2   |
|     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  + |
|      9       8      10       11       12           13       15       14     |
|     ------------------------------------------------------------------------|
|                           2   2                         2                   |
|     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d |
|      16       17       18          19       21       20      22       23    |
|     ------------------------------------------------------------------------|
|           2                                                                 |
|     + t  d )                                                                |
|        24                                                                   |
|                                                                             |
|o7 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17   2|
|-----------------------------------------------------------------------------|
| , t , t , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]                     |
|3   7   9   14   16   20   22   13   15   19   21                            |
+-----------------------------------------------------------------------------+
|i8 : J = groebnerStratum F                                                   |
|                                                                             |
|                                                                             |
|o8 = ideal (- t  + t   - t  t  , - t  - t  t  , - t   + t   + t t   - t  t   |
|               7    14    13 19     8    20 13     10    17    9 13    22 13 |
|     ------------------------------------------------------------------------|
|                         2                                                   |
|     - t t   - t  t   + t  t  , - t  t   + t  t  t  , - t   + t t   - t  t   |
|        7 15    16 19    13 21     16 21    13 15 21     11    9 14    16 20 |
|     ------------------------------------------------------------------------|
|                                                                             |
|     - t  t   - t t   + t  t  t  , t   + t t   - t  t   - t  t   + t  t  t  ,|
|        23 13    8 15    14 13 21   18    9 16    16 22    10 15    16 13 21 |
|     ------------------------------------------------------------------------|
|                                                                             |
|     - t   + t  t  - t  t   - t  t   - t  t   + t  t  t  , t  t  - t  t   -  |
|        12    17 9    23 16    24 13    11 15    17 13 21   18 9    24 16    |
|     ------------------------------------------------------------------------|
|                                           2              2        2         |
|     t  t   + t  t  t  , - t  - 2t  t   + t  t  , - t  - t   + t  t  , - t  -|
|      12 15    18 13 21     1     14 13    13 19     2    14    20 13     4  |
|     ------------------------------------------------------------------------|
|                                            2                                |
|     t t   - t  t   + t t   - 2t  t   + t  t   - t t   + t t  t   + t  t  t  |
|      7 16    14 16    3 13     17 13    22 13    1 15    7 13 15    14 13 15|
|     ------------------------------------------------------------------------|
|                    3                                            2           |
|     + t  t  t   - t  t  , t   - t t   - t  t   + t t  t   + t  t   +        |
|        16 13 19    13 21   18    9 16    17 15    9 13 15    14 15          |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     t  t  t   - t  t  t  , - t  + t t   - 2t  t   - t t   + t  t  t   +     |
|      16 13 21    13 15 21     5    3 14     17 14    8 16    16 20 13       |
|     ------------------------------------------------------------------------|
|         2             2                      2                              |
|     t  t   - t t   + t  t   + t t  t   - t  t  t  , t t   - t  t   - t  t   |
|      23 13    2 15    14 15    8 13 15    14 13 21   3 16    10 16    17 16 |
|     ------------------------------------------------------------------------|
|                                                                2            |
|     - t  t   + t  t  t   - t t   + t  t  t   + t  t  t   - t  t  t  , - t  +|
|        18 13    16 22 13    4 15    14 16 15    10 13 15    16 13 21     6  |
|     ------------------------------------------------------------------------|
|              2                                      2                       |
|     t t   - t   - t  t   - t  t   + t  t  t   + t  t   - t t   + t  t  t   +|
|      3 17    17    18 14    11 16    23 16 13    24 13    5 15    17 14 15  |
|     ------------------------------------------------------------------------|
|                     2                                                       |
|     t  t  t   - t  t  t  , t  t  - t  t   - t  t   + t  t  t   - t t   +    |
|      11 13 15    17 13 21   18 3    18 17    12 16    24 16 13    6 15      |
|     ------------------------------------------------------------------------|
|                                 2                                           |
|     t  t  t   + t  t  t   - t  t  t  , - t  - t  t   + t t   - t  t   +     |
|      18 14 15    12 13 15    18 13 21     8    20 13    7 19    14 19       |
|     ------------------------------------------------------------------------|
|         2                                                                   |
|     t  t  , t t   - t  t   + t  t  t  , - t   + t t  - t  t   - t  t   +    |
|      13 19   7 20    14 20    20 13 19     11    7 9    16 20    23 13      |
|     ------------------------------------------------------------------------|
|                                                              2              |
|     t  t  t   + t  t   - t  t   + t  t  t   + t t  t   + t  t   - t t   -   |
|      20 13 15    10 19    17 19    22 13 19    7 15 19    16 19    1 21     |
|     ------------------------------------------------------------------------|
|                 2                2                        2                 |
|     t t  t   - t  t  t  , t   + t  - t t   - t  t   + t  t   + t t  t   -   |
|      7 13 21    13 19 21   24    9    9 22    23 15    20 15    9 15 19     |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t t   + t  t   - t t  t   + t  t  t   - t  t  t  t  , t  t  + t t  -    |
|      3 21    10 21    7 15 21    16 19 21    13 15 19 21   23 7    8 9      |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t   + t  t   - t  t   - t t   + t  t  t   + t  t  t   + t  t  t   +  |
|      23 14    10 20    17 20    8 22    14 20 15    16 20 19    23 13 19    |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t t  t   - t t   - t t  t   - t  t  t  t  , - t   + t  t  - t  t   +    |
|      8 15 19    2 21    7 14 21    14 13 19 21     12    10 9    23 16      |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t  t   - t  t   + t  t  t   + t  t  t   - t t   - t t  t   -         |
|      16 20 15    18 19    16 22 19    10 15 19    4 21    7 16 21           |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t  t  t  , t  t   - t  t   + t  t  + t  t  - t  t   - t  t   +       |
|      16 13 19 21   10 23    17 23    24 7    11 9    18 20    11 22         |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t  t   + t  t  t   + t  t  t   + t  t  t   - t t   - t  t t   -      |
|      17 20 15    23 16 19    24 13 19    11 15 19    5 21    17 7 21        |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t  t  t  , t  t   - t  t   + t  t  - t  t   + t  t  t   + t  t  t   +|
|      17 13 19 21   24 10    18 23    12 9    12 22    18 20 15    24 16 19  |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t  t  t   - t t   - t  t t   - t  t  t  t  , - t  - t t   - t t   +     |
|      12 15 19    6 21    18 7 21    18 13 19 21     2    7 14    8 13       |
|     ------------------------------------------------------------------------|
|                                                                             |
|     t t   + t t  t  , - t t   + t t   + t t  t  , - t  + t t  - t  t  -     |
|      1 19    7 13 19     8 14    1 20    7 20 13     5    3 7    10 7       |
|     ------------------------------------------------------------------------|
|                                                         2                   |
|     t  t  - t t  - t t   + t t   - t  t   + t t  t   + t t   + t t  t   +   |
|      17 7    1 9    8 16    1 22    11 13    7 22 13    7 15    8 13 15     |
|     ------------------------------------------------------------------------|
|                                      2                                      |
|     t t   + t t  t   - t t  t   - t t  t  , t   - t  t  - t  t   + t t t   +|
|      4 19    7 16 19    1 13 21    7 13 21   12    10 9    11 15    7 9 15  |
|     ------------------------------------------------------------------------|
|        2                                                                    |
|     t t   + t t   + t t  t   - t t  t   - t t  t  t  , t t  - t t   - t t   |
|      8 15    4 21    7 16 21    1 15 21    7 13 15 21   3 8    8 10    8 17 |
|     ------------------------------------------------------------------------|
|                                                                             |
|     + t t   - t t  - t  t   + t t   + t t  t   + t  t t   + t t t   +       |
|        1 23    2 9    11 14    4 20    7 16 20    23 7 13    8 7 15         |
|     ------------------------------------------------------------------------|
|                                                        2                    |
|     t t  t   - t t  t   - t t  t  t  , - t  + t t   - t   - t  t  - t t  -  |
|      8 14 15    1 14 21    7 14 13 21     6    3 10    10    18 7    4 9    |
|     ------------------------------------------------------------------------|
|     t  t   + t t   + t t  t   + t  t t   + t t  t   - t t  t   -            |
|      11 16    4 22    7 16 22    10 7 15    8 16 15    1 16 21              |
|     ------------------------------------------------------------------------|
|     t t  t  t  , t  t  + t  t  - t  t  - t  t   - t  t   + t t   - t t  +   |
|      7 16 13 21   24 1    11 3    18 8    11 10    11 17    4 23    5 9     |
|     ------------------------------------------------------------------------|
|     t  t t   + t  t t   + t t  t   + t  t t   - t t  t   - t  t t  t  , -   |
|      23 7 16    24 7 13    8 17 15    11 7 15    1 17 21    17 7 13 21      |
|     ------------------------------------------------------------------------|
|     t  t   + t t   + t  t  - t  t   - t t  + t  t t   + t  t t   + t  t t   |
|      11 18    4 24    12 3    12 10    6 9    24 7 16    18 8 15    12 7 15 |
|     ------------------------------------------------------------------------|
|     - t  t t   - t  t t  t  )                                               |
|        18 1 21    18 7 13 21                                                |
|                                                                             |
|o8 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17   2|
|-----------------------------------------------------------------------------|
| , t , t , t  , t  , t  , t  , t  , t  , t  , t  ]                           |
|3   7   9   14   16   20   22   13   15   19   21                            |
+-----------------------------------------------------------------------------+
|i9 : compsJ = decompose J;                                                   |
+-----------------------------------------------------------------------------+
|i10 : compsJ = compsJ/trim;                                                  |
+-----------------------------------------------------------------------------+
|i11 : #compsJ == 2                                                           |
|                                                                             |
|o11 = true                                                                   |
+-----------------------------------------------------------------------------+
|i12 : compsJ/dim                                                             |
|                                                                             |
|o12 = {11, 8}                                                                |
|                                                                             |
|o12 : List                                                                   |
+-----------------------------------------------------------------------------+

There are 2 components.  We attempt to find a point on the first component

+-----------------------------------------------------------------------------+
|i13 : pt1 = randomPointOnRationalVariety compsJ_0                            |
|                                                                             |
|o13 = | 28 11 -25 -39 50 -28 -20 20 11 -24 43 37 -22 -8 -37 -29 -38 12 19 -29|
|      -----------------------------------------------------------------------|
|      19 -10 -16 -29 |                                                       |
|                                                                             |
|               1       24                                                    |
|o13 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i14 : F1 = sub(F, (vars S)|pt1)                                              |
|                                                                             |
|              2              2                              2                |
|o14 = ideal (a  + 11b*c - 39c  - 24a*d + 50b*d + 11c*d + 28d , a*b - 37b*c + |
|      -----------------------------------------------------------------------|
|         2                              2                   2                |
|      43c  - 29a*d + 37b*d - 28c*d - 25d , a*c + 19b*c - 38c  - 10a*d + 12b*d|
|      -----------------------------------------------------------------------|
|                   2   2              2                             2        |
|      - 22c*d - 20d , b  - 16b*c + 19c  - 29a*d - 29b*d - 8c*d + 20d )       |
|                                                                             |
|o14 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i15 : decompose F1                                                           |
|                                                                             |
|                                   2              2                    2     |
|o15 = {ideal (a + 19b - 38c + 2d, b  - 16b*c + 19c  + 17b*d + c*d - 23d ),   |
|      -----------------------------------------------------------------------|
|      ideal (c - 10d, b + 26d, a + 5d)}                                      |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+

We attempt to find a point on the second component in parameter space, and its
corresponding ideal.

+-----------------------------------------------------------------------------+
|i16 : pt2 = randomPointOnRationalVariety compsJ_1                            |
|                                                                             |
|o16 = | 19 45 9 15 -23 11 -36 -19 33 17 -7 -24 -47 50 20 42 -18 -39 21 39 34 |
|      -----------------------------------------------------------------------|
|      19 -13 0 |                                                             |
|                                                                             |
|               1       24                                                    |
|o16 : Matrix kk  <-- kk                                                      |
+-----------------------------------------------------------------------------+
|i17 : F2 = sub(F, (vars S)|pt2)                                              |
|                                                                             |
|              2              2                              2                |
|o17 = ideal (a  + 33b*c + 15c  + 17a*d - 23b*d + 45c*d + 19d , a*b + 20b*c - |
|      -----------------------------------------------------------------------|
|        2                             2                   2                  |
|      7c  + 42a*d - 24b*d + 11c*d + 9d , a*c + 34b*c - 18c  + 19a*d - 39b*d -|
|      -----------------------------------------------------------------------|
|                 2   2              2                      2                 |
|      47c*d - 36d , b  - 13b*c + 21c  + 39b*d + 50c*d - 19d )                |
|                                                                             |
|o17 : Ideal of S                                                             |
+-----------------------------------------------------------------------------+
|i18 : decompose F2                                                           |
|                                                                             |
|o18 = {ideal (b - 37c + 46d, a + 28c + 50d), ideal (b + 24c - 7d, a - 26c +  |
|      -----------------------------------------------------------------------|
|      5d)}                                                                   |
|                                                                             |
|o18 : List                                                                   |
+-----------------------------------------------------------------------------+

It turns out that this is the ideal of 2 skew lines, just not defined over this
field.

Caveat
======

This routine expects the input to represent an irreducible variety

See also
========

  * randomPointsOnRationalVariety
    -- find random points on a
    variety that can be detected to be rational

Ways to use this method:
========================

  * randomPointOnRationalVariety(Ideal)
    -- find a random point on a
    variety that can be detected to be rational
-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:939:0.

randomPointsOnRationalVariety(Ideal,ZZ) -- find random points on a variety that can be detected to be rational
**************************************************************************************************************

  * Function: randomPointsOnRationalVariety,
  * Usage:
        randomPointsOnRationalVariety(I, n)
        randomPointOnRationalVariety
  * Inputs:
      * I, an ideal (Macaulay2Doc), An ideal in a polynomial ring
        $S$ over a field, which defines a prime ideal
      * n, an integer (Macaulay2Doc), The number of points to
        generate
  * Outputs:
      * a list (Macaulay2Doc), A list of $n$ one row matrices over
        the base field of $S$, that are randomly chosen points on $I$.  null is
        returned in the case when the routine cannot determine if the variety
        is rational and irreducible.

Description
===========

+-----------------------------------------------------------------------------+
|i1 : kk = ZZ/101;                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = kk[a..f];                                                           |
+-----------------------------------------------------------------------------+
|i3 : I = minors(2, genericSymmetricMatrix(S, 3))                             |
|                                                                             |
|               2                                                  2          |
|o3 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - |
|     ------------------------------------------------------------------------|
|                                             2                               |
|     c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)                        |
|                                                                             |
|o3 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i4 : pts = randomPointsOnRationalVariety(I, 4)                               |
|                                                                             |
|o4 = {| 1 49 24 -23 -36 -30 |, | 23 -29 -29 19 19 19 |, | 38 -11 -10 -42 -29 |
|     ------------------------------------------------------------------------|
|     -8 |, | -37 -35 -22 -14 -29 -24 |}                                      |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+
|i5 : for p in pts list sub(I, p) == 0                                        |
|                                                                             |
|o5 = {true, true, true, true}                                                |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i6 : S = kk[a..d];                                                           |
+-----------------------------------------------------------------------------+
|i7 : F = groebnerFamily ideal"a2,ab,ac,b2"                                   |
|                                                                             |
|             2                      2                      2                 |
|o7 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +  |
|                  1       3       2      4       5       6           7       |
|     ------------------------------------------------------------------------|
|                2                         2                              2   |
|     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  + |
|      9       8      10       11       12           13       15       14     |
|     ------------------------------------------------------------------------|
|                           2   2                         2                   |
|     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d |
|      16       17       18          19       21       20      22       23    |
|     ------------------------------------------------------------------------|
|           2                                                                 |
|     + t  d )                                                                |
|        24                                                                   |
|                                                                             |
|o7 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17   2|
|-----------------------------------------------------------------------------|
| , t , t , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]                     |
|3   7   9   14   16   20   22   13   15   19   21                            |
+-----------------------------------------------------------------------------+
|i8 : J = groebnerStratum F;                                                  |
|                                                                             |
|o8 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  ,   |
|                  6   5   12   2   4   11   18   24   1   3   8   10   17    |
|-----------------------------------------------------------------------------|
|t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ]                         |
| 23   7   9   14   16   20   22   13   15   19   21                          |
+-----------------------------------------------------------------------------+
|i9 : compsJ = decompose J;                                                   |
+-----------------------------------------------------------------------------+
|i10 : compsJ = compsJ/trim;                                                  |
+-----------------------------------------------------------------------------+
|i11 : #compsJ == 2                                                           |
|                                                                             |
|o11 = true                                                                   |
+-----------------------------------------------------------------------------+
|i12 : compsJ/dim                                                             |
|                                                                             |
|o12 = {11, 8}                                                                |
|                                                                             |
|o12 : List                                                                   |
+-----------------------------------------------------------------------------+

There are 2 components.  We attempt to find points on each of these two
components. We are successful.  This indicates that the corresponding varieties
are both rational. Also, if we can find one point, we can find as many as we
want.

+-----------------------------------------------------------------------------+
|i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)                    |
|                                                                             |
|      +----------------------------------------------------------------------|
|o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -1|
|      +----------------------------------------------------------------------|
|      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -|
|      +----------------------------------------------------------------------|
|      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17|
|      +----------------------------------------------------------------------|
|      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -|
|      +----------------------------------------------------------------------|
|      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -3|
|      +----------------------------------------------------------------------|
|      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -2|
|      +----------------------------------------------------------------------|
|      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 |
|      +----------------------------------------------------------------------|
|      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -4|
|      +----------------------------------------------------------------------|
|      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 3|
|      +----------------------------------------------------------------------|
|      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -|
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|---------------+                                                             |
|8 -43 21 -38 | |                                                             |
|---------------+                                                             |
|34 -47 38 -15 ||                                                             |
|---------------+                                                             |
| 48 7 47 |     |                                                             |
|---------------+                                                             |
|3 -38 36 |     |                                                             |
|---------------+                                                             |
|7 -10 -7 22 |  |                                                             |
|---------------+                                                             |
|0 -30 27 30 |  |                                                             |
|---------------+                                                             |
|17 44 0 -48 |  |                                                             |
|---------------+                                                             |
|9 -8 -39 -39 | |                                                             |
|---------------+                                                             |
|5 -3 43 |      |                                                             |
|---------------+                                                             |
|41 -13 40 -9 | |                                                             |
|---------------+                                                             |
+-----------------------------------------------------------------------------+
|i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)                    |
|                                                                             |
|      +----------------------------------------------------------------------|
|o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31|
|      +----------------------------------------------------------------------|
|      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 |
|      +----------------------------------------------------------------------|
|      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10|
|      +----------------------------------------------------------------------|
|      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 1|
|      +----------------------------------------------------------------------|
|      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 4|
|      +----------------------------------------------------------------------|
|      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 2|
|      +----------------------------------------------------------------------|
|      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 |
|      +----------------------------------------------------------------------|
|      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -3|
|      +----------------------------------------------------------------------|
|      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 |
|      +----------------------------------------------------------------------|
|      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------+                                                           |
| -35 -47 0 |     |                                                           |
|-----------------+                                                           |
|-48 0 |          |                                                           |
|-----------------+                                                           |
| -22 -18 0 |     |                                                           |
|-----------------+                                                           |
|3 0 |            |                                                           |
|-----------------+                                                           |
|2 30 0 |         |                                                           |
|-----------------+                                                           |
|7 -18 18 0 |     |                                                           |
|-----------------+                                                           |
|-39 19 20 -37 0 ||                                                           |
|-----------------+                                                           |
|3 -9 -28 0 |     |                                                           |
|-----------------+                                                           |
|42 -28 26 0 |    |                                                           |
|-----------------+                                                           |
|-13 4 0 |        |                                                           |
|-----------------+                                                           |
+-----------------------------------------------------------------------------+

Caveat
======

This routine expects the input to represent an irreducible variety

See also
========

  * randomPointOnRationalVariety(Ideal)
    -- find a random point on a
    variety that can be detected to be rational

Ways to use this method:
========================

  * randomPointsOnRationalVariety(Ideal,ZZ)
    -- find random points on a
    variety that can be detected to be rational
-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:883:0.

smallerMonomials -- returns the standard monomials smaller but of the same degree as given monomial(s)
******************************************************************************************************

  * Usage:
        L = smallerMonomials M
        L = smallerMonomials(M, m)
  * Inputs:
      * M, an ideal (Macaulay2Doc), $M$ should be a monomial ideal
        (an ideal generated by monomials)
      * m, a ring element (Macaulay2Doc), optional,
  * Outputs:
      * L, a list (Macaulay2Doc), a list of lists: for each
        generator $m$ of $M$, the list of all monomials of the same degree as
        $m$, not in the monomial ideal and smaller than that generator in the
        term order of the ambient ring.  If instead $m$ is given, the list of
        the standard monomials of the same degree, smaller than $m$, is
        returned.

Description
===========

Inputting an ideal $M$ returns the smaller monomials of each of the given
generators of the ideal.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a..d];                                                     |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, b^2, a*b*c);                                            |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : L1 = smallerMonomials M                                                 |
|                                                                             |
|                       2                  2               2                  |
|o3 = {{a*b, a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d,  |
|     ------------------------------------------------------------------------|
|      2       2     2   3                        2      2     2     2   3    |
|     d }, {a*c , b*c , c , a*b*d, a*c*d, b*c*d, c d, a*d , b*d , c*d , d }}  |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : smallerMonomials(M, b^2)                                                |
|                                                                             |
|                 2                  2                                        |
|o4 = {a*c, b*c, c , a*d, b*d, c*d, d }                                       |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * tailMonomials -- find tail monomials
  * standardMonomials -- computes standard monomials

Ways to use smallerMonomials:
=============================

  * "smallerMonomials(Ideal)"
  * "smallerMonomials(Ideal,RingElement)"

For the programmer
==================

The object smallerMonomials, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:524:0.

standardMonomials -- computes standard monomials
************************************************

  * Usage:
        L = standardMonomials M
        L = standardMonomials(d, M)
  * Inputs:
      * M, an ideal (Macaulay2Doc), M should be a monomial ideal
      * d, a list (Macaulay2Doc), a degree
  * Outputs:
      * L, a list (Macaulay2Doc), L is a list of lists of standard
        monomials for the ideal $M$, one for each generator of $M$

Description
===========

A monomial $m$ is standard with respect to a monomial ideal $M$ and a generator
$g$ of $M$ if $m$ is of the same degree as $g$ but is not an element of $M$.

Inputting an ideal $M$ returns the standard monomials of each of the given
generators of the ideal.

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a..d];                                                     |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, a*b, b^3, a*c);                                         |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : L1 = standardMonomials M                                                |
|                                                                             |
|        2        2                  2     2        2                  2      |
|o3 = {{b , b*c, c , a*d, b*d, c*d, d }, {b , b*c, c , a*d, b*d, c*d, d },    |
|     ------------------------------------------------------------------------|
|       2      2   3   2           2      2     2     2   3     2        2    |
|     {b c, b*c , c , b d, b*c*d, c d, a*d , b*d , c*d , d }, {b , b*c, c ,   |
|     ------------------------------------------------------------------------|
|                     2                                                       |
|     a*d, b*d, c*d, d }}                                                     |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : standardMonomials({3}, M)                                               |
|                                                                             |
|       2      2   3   2           2      2     2     2   3                   |
|o4 = {b c, b*c , c , b d, b*c*d, c d, a*d , b*d , c*d , d }                  |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

Inputting an integer $d$ (or degree $d$) and an ideal gives the standard
monomials for the specified ideal in degree $d$.

+-------------------------------------+
|i5 : standardMonomials(2, M)         |
|                                     |
|       2        2                  2 |
|o5 = {b , b*c, c , a*d, b*d, c*d, d }|
|                                     |
|o5 : List                            |
+-------------------------------------+

See also
========

  * tailMonomials -- find tail monomials
  * smallerMonomials -- returns the standard monomials
    smaller but of the same degree as given monomial(s)

Ways to use standardMonomials:
==============================

  * "standardMonomials(Ideal)"
  * "standardMonomials(List,Ideal)"
  * "standardMonomials(ZZ,Ideal)"

For the programmer
==================

The object standardMonomials, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:488:0.

tailMonomials -- find tail monomials
************************************

  * Usage:
        L = tailMonomials M
        L = tailMonomials(M, m)
  * Inputs:
      * M, an ideal (Macaulay2Doc), $M$ should be a monomial ideal
        (an ideal generated by monomials)
      * m, a ring element (Macaulay2Doc), optional, only
        return a single list of the tail monomials for this monomial
  * Optional inputs (Macaulay2Doc),:
      * AllStandard => a Boolean value (Macaulay2Doc), default
        value false, which monomials should be considered tail monomials of a
        monomial $m$: either all standard monomials of a given degree, or all
        monomials smaller than $m$ in the given term order (but still of the
        same degree)
  * Outputs:
      * L, a list (Macaulay2Doc), a list of lists: for each
        generator $m$ of $M$, the list of all tail monomials If instead $m$ is
        given, the list of the tail monomials of $m$ is returned

Description
===========

Inputting an ideal $M$ generated by monomials returns a list of lists of tail
monomials for each generator of $M$ (in the same order).

+-----------------------------------------------------------------------------+
|i1 : R = ZZ/32003[a..d];                                                     |
+-----------------------------------------------------------------------------+
|i2 : M = ideal (a^2, b^2, a*b*c);                                            |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : tailMonomials M                                                         |
|                                                                             |
|                       2                  2               2                  |
|o3 = {{a*b, a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d,  |
|     ------------------------------------------------------------------------|
|      2       2     2   3                        2      2     2     2   3    |
|     d }, {a*c , b*c , c , a*b*d, a*c*d, b*c*d, c d, a*d , b*d , c*d , d }}  |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : tailMonomials(M, AllStandard => true)                                   |
|                                                                             |
|                       2                  2                    2             |
|o4 = {{a*b, a*c, b*c, c , a*d, b*d, c*d, d }, {a*b, a*c, b*c, c , a*d, b*d,  |
|     ------------------------------------------------------------------------|
|           2       2     2   3                        2      2     2     2   |
|     c*d, d }, {a*c , b*c , c , a*b*d, a*c*d, b*c*d, c d, a*d , b*d , c*d ,  |
|     ------------------------------------------------------------------------|
|      3                                                                      |
|     d }}                                                                    |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+
|i5 : tailMonomials(M, b^2)                                                   |
|                                                                             |
|                 2                  2                                        |
|o5 = {a*c, b*c, c , a*d, b*d, c*d, d }                                       |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+
|i6 : tailMonomials(M, b^2, AllStandard=>true)                                |
|                                                                             |
|                      2                  2                                   |
|o6 = {a*b, a*c, b*c, c , a*d, b*d, c*d, d }                                  |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * standardMonomials -- computes standard monomials
  * smallerMonomials -- returns the standard monomials
    smaller but of the same degree as given monomial(s)

Ways to use tailMonomials:
==========================

  * "tailMonomials(Ideal)"
  * "tailMonomials(Ideal,RingElement)"

For the programmer
==================

The object tailMonomials, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GroebnerStrata.m2:565:0.