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

Synopsis

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

Ways to use groebnerStratum:

For the programmer

The object groebnerStratum is a method function with options.