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

ExampleSystems -- examples of polynomial systems
************************************************

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

This package is a database for examples of polynomial systems in Macaulay2,
including standard systems of interest in various applications, such as
engineering, chemistry, robotics, and economics.

References:

  * http://homepages.math.uic.edu/~jan/demo.html
  * http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/node1.html
  * http://symbolicdata.org/XMLResources/IntPS/
  * http://www.cecm.sfu.ca/~rpearcea/polsys.txt

Authors
=======

  * Anton Leykin <leykin@math.gatech.edu>
  * Justin Chen <justin.chen@math.gatech.edu>
  * Kelly Maluccio <kmaluccio@math.tamu.edu>
  * Leah Gold <L.Gold33@csuohio.edu>

Version
=======

This documentation describes version 0.5 of ExampleSystems.

Citation
========

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

+----------------------------------------------------------------------------+
|@misc{ExampleSystemsSource,                                                 |
|  title = {{ExampleSystems: database of polynomial systems. Version~0.5}},  |
|  author = {Anton Leykin and Justin Chen and Kelly Maluccio and Leah Gold}, |
|  howpublished = {A \emph{Macaulay2} package available at                   |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}|
|}                                                                           |
|                                                                            |
+----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * bellido -- Bellido system
      * boon -- a neurophysiology problem posted by Sjirk Boon
      * butcher -- Butcher's problem
      * camera1s -- a system for camera displacement between
        two positions, with scaled first frame.
      * caprasse -- the system caprasse of the PoSSo test
        suite
      * cassou -- the system of Pierrette Cassou-Nogu`es
      * chemequ -- chemical equilibrium of hydrocarbon
        combustion
      * cohn3 -- modular equations for special algebraic number
        fields
      * comb3000s -- a combustion chemistry example for a
        temperature of 3000 degrees
      * cyclic -- an example of a square polynomial system
      * dipole2 -- Dipole2 system
      * eco8 -- an 8-dimensional economics problem
      * geneig -- generalized eigenvalue problem
      * heart -- an 8-dimensional economics problem
      * ipp -- six-revolute-joint problem of mechanics
      * katsura -- an example of a 0-dimensional square
        polynomial system
      * ku10 -- a 10-dimensional system of Ku
      * lorentz -- equilibrium points of a 4-dimensional
        Lorentz attractor
      * lumped -- lumped parameter chemically reacting system
      * noon5 -- A neural network modeled by an adaptive
        Lotka-Volterra system with 5 variables
      * proddeco -- system with a product-decomposition
        structure
      * puma -- hand position and orientation of PUMA robot
      * quadgrid -- interpolating quadrature formula for
        function defined on a grid
      * rabmo -- optimal multi-dimensional quadrature formulas
      * randomGeneralizedEigenvalueProblem
        -- an example of a 0-dimensional
        square polynomial system
      * randomSystem -- an example of a 0-dimensional
        square polynomial system
      * rbpl24s -- parallel robot with 24 real solutions, after
        variable scaling
      * reimer5 -- 5-dimensional Reimer system
      * rose -- the system Rose, a general economic equilibrium
        model
      * sendra -- the system sendra
      * trinks -- system of Trinks from the PoSSo test suite
      * virasoro -- the construction of Virasoro algebras
      * wood -- system derived from optimizing the Wood function
      * wright -- system of A.H. Wright
  * Methods
      * "bellido(Ring)" -- see bellido, -- Bellido system
      * "boon(Ring)" -- see boon, -- a neurophysiology problem
        posted by Sjirk Boon
      * "butcher(Ring)" -- see butcher, -- Butcher's problem
      * "camera1s(Ring)" -- see camera1s, -- a system for
        camera displacement between two positions, with scaled first frame.
      * "caprasse(Ring)" -- see caprasse, -- the system
        caprasse of the PoSSo test suite
      * "cassou(Ring)" -- see cassou, -- the system of Pierrette
        Cassou-Nogu`es
      * "chemequ(Ring)" -- see chemequ, -- chemical equilibrium
        of hydrocarbon combustion
      * "cohn3(Ring)" -- see cohn3, -- modular equations for
        special algebraic number fields
      * "comb3000s(Ring)" -- see comb3000s, -- a combustion
        chemistry example for a temperature of 3000 degrees
      * "cyclic(ZZ,Ring)" -- see cyclic, -- an example of a
        square polynomial system
      * "dipole2(Ring)" -- see dipole2, -- Dipole2 system
      * "eco8(Ring)" -- see eco8, -- an 8-dimensional economics
        problem
      * "geneig(Ring)" -- see geneig, -- generalized eigenvalue
        problem
      * "heart(Ring)" -- see heart, -- an 8-dimensional economics
        problem
      * "ipp(Ring)" -- see ipp, -- six-revolute-joint problem of
        mechanics
      * "katsura(ZZ,Ring)" -- see katsura, -- an example of a
        0-dimensional square polynomial system
      * "ku10(Ring)" -- see ku10, -- a 10-dimensional system of Ku
      * "lorentz(Ring)" -- see lorentz, -- equilibrium points of
        a 4-dimensional Lorentz attractor
      * "lumped(Ring)" -- see lumped, -- lumped parameter
        chemically reacting system
      * "noon5(Ring)" -- see noon5, -- A neural network modeled by
        an adaptive Lotka-Volterra system with 5 variables
      * "proddeco(Ring)" -- see proddeco, -- system with a
        product-decomposition structure
      * "puma(Ring)" -- see puma, -- hand position and orientation
        of PUMA robot
      * "quadgrid(Ring)" -- see quadgrid, -- interpolating
        quadrature formula for function defined on a grid
      * "rabmo(Ring)" -- see rabmo, -- optimal multi-dimensional
        quadrature formulas
      * "randomGeneralizedEigenvalueProblem(ZZ)" -- see
        randomGeneralizedEigenvalueProblem,
        -- an example of a 0-dimensional square polynomial system
      * "randomSystem(ZZ,ZZ,Ring)" -- see randomSystem, --
        an example of a 0-dimensional square polynomial system
      * "rbpl24s(Ring)" -- see rbpl24s, -- parallel robot with
        24 real solutions, after variable scaling
      * "reimer5(Ring)" -- see reimer5, -- 5-dimensional Reimer
        system
      * "rose(Ring)" -- see rose, -- the system Rose, a general
        economic equilibrium model
      * "sendra(Ring)" -- see sendra, -- the system sendra
      * "trinks(Ring)" -- see trinks, -- system of Trinks from
        the PoSSo test suite
      * "virasoro(Ring)" -- see virasoro, -- the construction
        of Virasoro algebras
      * "wood(Ring)" -- see wood, -- system derived from optimizing
        the Wood function
      * "wright(Ring)" -- see wright, -- system of A.H. Wright

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

The object ExampleSystems, is a package (Macaulay2Doc),
defined in ExampleSystems.m2, with auxiliary files in
ExampleSystems/.

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

The source of this document is in ExampleSystems.m2:86:0.

bellido -- Bellido system
*************************

  * Usage:
        bellido(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-8250U CPU at 1.60GHz.

There were 40 solutions found in 0.546875 seconds (with a Bezout bound of 64).

Reference: Bellido A.M. "Construction of iteration functions for the
simultaneous computation of the solutions of equations and algebraic systems."
Numerical Algorithms, 6(3-4):315-351, 1992.

See also: http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/node1.html#
SECTION00012000000000000000

+-----------------------------------------------------------------------------+
|i1 : bellido(QQ)                                                             |
|                                                                             |
|       2    2    2               2    2    2               2    2    2       |
|o1 = {z  + z  + z  - 12z  - 68, z  + z  + z  - 12z  - 68, z  + z  + z  - 24z |
|       1    2    3      1        4    5    6      5        7    8    9      8|
|     ------------------------------------------------------------------------|
|     - 12z  + 100, z z  + z z  + z z  - 6z  - 6z  - 52, z z  + z z  + z z  - |
|          9         1 4    2 5    3 6     1     5        1 7    2 8    3 9   |
|     ------------------------------------------------------------------------|
|     6z  - 12z  - 6z  + 64, z z  + z z  + z z  - 6z  - 12z  - 6z  + 32, 2z  +|
|       1      8     9        4 7    5 8    6 9     5      8     9         2  |
|     ------------------------------------------------------------------------|
|     2z  - z  - z  - 2z  - z  - z  + 18, z  + z  + 2z  + 2z  + 2z  - 2z  + z |
|       3    4    5     6    7    9        1    2     3     4     6     7    8|
|     ------------------------------------------------------------------------|
|     - z  - 38, z  + z  - 2z  + z  - z  + 2z  - 2z  + 8}                     |
|        9        1    3     4    5    6     7     8                          |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use bellido:
====================

  * "bellido(Ring)"

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

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

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

The source of this document is in ExampleSystems/bellido.m2:50:0.

boon -- a neurophysiology problem posted by Sjirk Boon
******************************************************

  * Usage:
        boon(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 8 solutions found in 10.1037 seconds (with a Bezout bound of 1024).

Reference: P. Van Hentenryck, D. McAllester and D. Kapur: "Solving Polynomial
Systems Using a Branch and Prune Approach" SIAM J. Numerical Analysis, Vol. 34,
No. 2, pp 797-827, 1997.

See also: post to sci.math.num-analysis by Sjirk Boon

See also: http://homepages.math.uic.edu/~jan/Demo/boon.html

+-----------------------------------------------------------------------------+
|i1 : boon(QQ)                                                                |
|                                                                             |
|       2    2       2    2       3      3     6   3      3     6     2       |
|o1 = {s  + g  - 1, s  + g  - 1, g C  + g C  - -, s C  + s C  - -, s g C  +   |
|       1    1       2    2       1 1    2 2   5   1 1    2 2   5   1 1 1     |
|     ------------------------------------------------------------------------|
|        2      7   2        2        7                                       |
|     s g C  - --, s g C  + s g C  - --}                                      |
|      2 2 2   10   1 1 1    2 2 2   10                                       |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use boon:
=================

  * "boon(Ring)"

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

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

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

The source of this document is in ExampleSystems/boon.m2:49:0.

butcher -- Butcher's problem
****************************

  * Usage:
        butcher(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), a list of the polynomials in this
        example

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 368 solutions found in 82.589 seconds (with a Bezout bound of 4608).

References:

W. Boege, R. Gebauer, and H. Kredel: "Some examples for solving systems of
algebraic equations by calculating Groebner bases", J. Symbolic Computation,
2:83-98, 1986.

C. Butcher: "An application of the Runge-Kutta space". BIT, 24, pages 425–440,
1984.

This example is from the POSSO test suite available by anonymous ftp from the
site gauss.dm.unipi.it, from the directory pub/posso.

See also: http://homepages.math.uic.edu/~jan/Demo/butcher.html

+-----------------------------------------------------------------------------+
|i1 : butcher(QQ)                                                             |
|                                                                             |
|                         2   1    1     2      2      2    3    2   1    4   |
|o1 = {z*u + y*v + t*w - w  - -w - -, z*u  + y*v  - t*w  + w  + w  - -t + -w, |
|                             2    2                                 3    3   |
|     ------------------------------------------------------------------------|
|          2    3           1       2   1    2      3      3      3    4   3 3|
|     - t*w  + w  + z*v*x - -t*w + w  - -t + -w, z*u  + y*v  + t*w  - w  - -w |
|                           2           6    3                             2  |
|     ------------------------------------------------------------------------|
|             5 2   1    1     3    4             1   2   3 3   1      7 2    |
|     + t*w - -w  - -w - -, t*w  - w  + z*u*v*x + -t*w  - -w  + -t*w - -w  -  |
|             2     4    4                        2       2     2      4      |
|     ------------------------------------------------------------------------|
|     3    1     3    4      2       2   3 3   2      7 2    1     1       3  |
|     -w - -, t*w  - w  + z*v x + t*w  - -w  + -t*w - -w  - --w - --, - t*w  +|
|     8    8                             2     3      6     12    12          |
|     ------------------------------------------------------------------------|
|      4      2   3 3   1      13 2    7     1                                |
|     w  - t*w  + -w  - -t*w + --w  + --w + --}                               |
|                 2     3      12     24    24                                |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use butcher:
====================

  * "butcher(Ring)"

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

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

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

The source of this document is in ExampleSystems/butcher.m2:55:0.

camera1s -- a system for camera displacement between two positions, with scaled first frame.
********************************************************************************************

  * Usage:
        camera1s(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 20 solutions found in 0.2575 seconds (with a Bezout bound of 64).

Reference:

Ioannis Z. Emiris. "Sparse Elimination and Application in Kinematics". PhD
Thesis, Computer Science, University of California at Berkeley, 1994.

Ioannis Z. Emiris. "A general Solver Based on Sparse Resultants: Numerical
Issues and Kinematic Applications", INRIA Rapport de Recherche no 3110, January
1997, 29 pages. Available via anonymous ftp to ftp.inria.fr

See also: http://homepages.math.uic.edu/~jan/Demo/camera1s.html

+-----------------------------------------------------------------------------+
|i1 : camera1s(RR_101)                                                        |
|                                                                             |
|o1 = {- d q  - d q  - d q  + 1, - 3.6d q  + 4.1d q  + 2d q  + 4.1d q  +      |
|         1 1    2 2    3 3            1 1       2 1     3 1       1 2        |
|     ------------------------------------------------------------------------|
|     1.8d q  + 3.7d q  + 2d q  + 3.7d q  - 4d q  + .1d  - .2d  + .3d  + .1q  |
|         2 2       3 2     1 3       2 3     3 3      1      2      3      1 |
|     ------------------------------------------------------------------------|
|     - .2q  + .3q  + 5.8, - 2.1408d q  - 3.99879d q  + 3.71599d q  -         |
|          2      3                 1 1           2 1           3 1           |
|     ------------------------------------------------------------------------|
|     3.99879d q  - 1.5752d q  - 3.99879d q  + 3.71599d q  - 3.99879d q  -    |
|             1 2          2 2           3 2           1 3           2 3      |
|     ------------------------------------------------------------------------|
|     2.1408d q  - .2828d  + .2828d  - .2828q  + .2828q  + 5.85679, .3464d q  |
|            3 3         1         3         1         3                  1 1 |
|     ------------------------------------------------------------------------|
|     + .1732d q  - 5.99965d q  + .1732d q  - 5.99965d q  - .1732d q  -       |
|             2 1           3 1         1 2           2 2         3 2         |
|     ------------------------------------------------------------------------|
|     5.99965d q  - .1732d q  - .3464d q  - .1732d  + .3464d  - .1732d  -     |
|             1 3         2 3         3 3         1         2         3       |
|     ------------------------------------------------------------------------|
|     .1732q  + .3464q  - .1732q  + 5.99965, - 5701.3d q  - 2.9d q  +         |
|           1         2         3                     1 1       2 1           |
|     ------------------------------------------------------------------------|
|     3796.7d q  - 2.9d q  - 5698.7d q  + 1897.3d q  + 3796.7d q  + 1897.3d q |
|            3 1       1 2          2 2          3 2          1 3          2 3|
|     ------------------------------------------------------------------------|
|     + 5703.1d q  - 1902.7d  + 3803.3d  + .7d  - 1902.7q  + 3803.3q  + .7q  +|
|              3 3          1          2      3          1          2      3  |
|     ------------------------------------------------------------------------|
|     5696.9, - 6.8d q  - 3.2d q  + 1.3d q  - 3.2d q  - 4.8d q  - .7d q  +    |
|                   1 1       2 1       3 1       1 2       2 2      3 2      |
|     ------------------------------------------------------------------------|
|     1.3d q  - .7d q  + 9d q  + 5.1d  - 7.1d  - d  + 5.1q  - 7.1q  - q  +    |
|         1 3      2 3     3 3       1       2    3       1       2    3      |
|     ------------------------------------------------------------------------|
|     2.6}                                                                    |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use camera1s:
=====================

  * "camera1s(Ring)"

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

The object camera1s, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/camera1s.m2:55:0.

caprasse -- the system caprasse of the PoSSo test suite
*******************************************************

  * Usage:
        caprasse(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 64 solutions found in 0.525506 seconds (with a Bezout bound of 144).

Reference:

The PoSSo test suite.

See also: http://homepages.math.uic.edu/~jan/Demo/caprasse.html

+-----------------------------------------------------------------------------+
|i1 : caprasse(QQ)                                                            |
|                                                                             |
|       2                        3        2      2        3      2      2     |
|o1 = {y z + 2x*y*t - 2x - z, - x z + 4x*y z + 4x y*t + 2y t + 4x  - 10y  +   |
|     ------------------------------------------------------------------------|
|                                   2                3       2          2     |
|     4x*z - 10y*t + 2, 2y*z*t + x*t  - x - 2z, - x*z  + 4y*z t + 4x*z*t  +   |
|     ------------------------------------------------------------------------|
|         3            2              2                                       |
|     2y*t  + 4x*z + 4z  - 10y*t - 10t  + 2}                                  |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use caprasse:
=====================

  * "caprasse(Ring)"

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

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

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

The source of this document is in ExampleSystems/caprasse.m2:45:0.

cassou -- the system of Pierrette Cassou-Nogu`es
************************************************

  * Usage:
        cassou(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz

There were 2 solutions found in 39.0405 seconds (with a Bezout bound of 1344).

Reference: Obtained by electronic mail by Carlo Traverso. See the POSSO test
suite.

T.Y. Li, Tianjun Wang, Xiaoshen Wang: "Random Product Homotopy with Minimal BKK
Bound", in "The Mathematics of Numerical Analysis", Edited by Renegar, J. and
Shub, M. and Smale, S., Lectures in Applied Mathematics vol 32, 1996.
Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics, Park City,
Utah, July 17-August 11, 1995, Park City, Utah".

See also: http://homepages.math.uic.edu/~jan/Demo/cassou.html

+-----------------------------------------------------------------------------+
|i1 : cassou(RR_53)                                                           |
|                                                                             |
|        4 3      4 2       4   2     4 3     2 2       2           2 2       |
|o1 = {6b c  + 21b c d + 15b c*d  + 9b d  - 8b c e - 28b c*d*e + 36b d e -    |
|     ------------------------------------------------------------------------|
|         2        2           4 4      4 3       4 2 2      4   3      2 3   |
|     144b c - 648b d - 120, 9b c  + 30b c d + 39b c d  + 18b c*d  - 24b c e -|
|     ------------------------------------------------------------------------|
|        2 2         2   2       2 3        2 2       2          2 2      2 2 |
|     16b c d*e + 16b c*d e + 24b d e - 432b c  - 720b c*d - 432b d  + 16c e  |
|     ------------------------------------------------------------------------|
|              2      2 2                                       2 3           |
|     - 32c*d*e  + 16d e  + 576c*e - 576d*e - 240c + 5184, - 15b c e +        |
|     ------------------------------------------------------------------------|
|        2 2         2 2       2          2 2      2 2          2      2 2    |
|     15b c d*e - 81b c  + 216b c*d - 162b d  + 40c e  - 80c*d*e  + 40d e  +  |
|     ------------------------------------------------------------------------|
|                                   2 2     2        2 2                      |
|     1008c*e - 1008d*e + 5184, - 4b c  + 4b c*d - 3b d  + 22c*e - 22d*e +    |
|     ------------------------------------------------------------------------|
|     261}                                                                    |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use cassou:
===================

  * "cassou(Ring)"

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

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

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

The source of this document is in ExampleSystems/cassou.m2:53:0.

chemequ -- chemical equilibrium of hydrocarbon combustion
*********************************************************

  * Usage:
        chemequ(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 6 solutions found in 1.24877 seconds (with a Bezout bound of 108).

Reference: Keith Meintjes and Alexander P. Morgan, "Chemical equilibrium
systems as numerical test problems", ACM Toms, Vol 16, No 2, 143-151, 1990.

See also: http://homepages.math.uic.edu/~jan/Demo/chemequ.html

+-----------------------------------------------------------------------------+
|i1 : chemequ(CC_53)                                                          |
|                                                                             |
|                          2                      2                           |
|o1 = {y y  + y  - 3y , y y  + 2y y  + .000001923y  + .00054518y y  +         |
|       1 2    1     5   2 3     1 2              2             2 3           |
|     ------------------------------------------------------------------------|
|                                                   2                        2|
|     .000034074y y  + y  + 4.4975e-7y  - 10y , 2y y  + .00054518y y  + .386y |
|                2 4    1             2      5    2 3             2 3        3|
|     ------------------------------------------------------------------------|
|                                             2            2                  |
|     + .00041062y  - 8y , .000034074y y  + 2y  - 40y , y y  + y y  +         |
|                 3     5             2 4     4      5   2 3    1 2           |
|     ------------------------------------------------------------------------|
|              2                        2                     2               |
|     9.615e-7y  + .00054518y y  + .193y  + .000034074y y  + y  + y  +        |
|              2             2 3        3              2 4    4    1          |
|     ------------------------------------------------------------------------|
|     4.4975e-7y  + .00041062y  - 1}                                          |
|               2             3                                               |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use chemequ:
====================

  * "chemequ(Ring)"

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

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

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

The source of this document is in ExampleSystems/chemequ.m2:47:0.

cohn3 -- modular equations for special algebraic number fields
**************************************************************

  * Usage:
        cohn3(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 72 solutions found in 13.8985 seconds (with a Bezout bound of 1080).

Reference: See the PoSSo test suite. Andre' Galligo and Carlo Traverso.
"Practical Determination of the dimension of an algebraic variety", in E.
Kaltofen and S.M. Watt, Eds "Computers and Mathematics", pages 46-52, 1989.

H. Cohn. "An explicit modular equation in two variables and Hilbert's Twelfth
problem", Math. of Comp. 38, pp. 227-236, 1982.

H. Cohn, J. Deutch. "An explicit modular equation in two variables for
Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.

See also: http://homepages.math.uic.edu/~jan/Demo/cohn3.html

+-----------------------------------------------------------------------------+
|i1 : F = cohn3(QQ)                                                           |
|                                                                             |
|         3 2     2 2     2   2       2 2       2            2           2    |
|o1 = {- x y  + 2x y z - x y*z  - 144x y  - 207x y*z + 288x*y z + 78x*y*z  +  |
|     ------------------------------------------------------------------------|
|        3        2           2                     2                         |
|     x*z  - 3456x y - 5184x*y  - 9504x*y*z - 432x*z  - 248832x*y + 62208x*z -|
|     ------------------------------------------------------------------------|
|                  3   2    2 2 2     3        2 2       3 2      2   2       |
|     2985984x, - x z*t  + x z t  - 6x z*t + 4x z t + 32x t  - 72x z*t  -     |
|     ------------------------------------------------------------------------|
|          2 2    3 2     3        2            2             2       2 2     |
|     87x*z t  - z t  - 8x z - 432x z*t - 414x*z t + 2592x*z*t  + 864z t  -   |
|     ------------------------------------------------------------------------|
|          2                      2             2                             |
|     1728x z - 20736x*z*t + 3456z t - 186624z*t  - 124416x*z - 1492992z*t -  |
|     ------------------------------------------------------------------------|
|                2   3       2 3    3 3     2   2        2 2     3 2          |
|     2985984z, x y*t  - 2x*y t  + y t  + 8x y*t  - 12x*y t  + 4y t  -        |
|     ------------------------------------------------------------------------|
|            3      2 3      2           2            2      2 2         3    |
|     24x*y*t  + 24y t  + 20x y*t - 20x*y t - 160x*y*t  + 96y t  + 128x*t  +  |
|     ------------------------------------------------------------------------|
|        2                     2                                 3 3    2   3 |
|     16x y + 96x*y*t + 2304x*t  + 1152x*y + 13824x*t + 27648x, y t  - y z*t  |
|     ------------------------------------------------------------------------|
|         3 2     2   2      2 3          3       2 2           2     2 2     |
|     + 4y t  - 2y z*t  + 72y t  + 71y*z*t  + 288y t  + 360y*z*t  + 6z t  +   |
|     ------------------------------------------------------------------------|
|            3         3                2           2          2         3    |
|     1728y*t  - 464z*t  + 432y*z*t + 8z t + 6912y*t  - 4320z*t  + 13824t  +  |
|     ------------------------------------------------------------------------|
|      2                    2                                                 |
|     z  - 13824z*t + 55296t  - 13824z}                                       |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use cohn3:
==================

  * "cohn3(Ring)"

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

The object cohn3, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/cohn3.m2:55:0.

comb3000s -- a combustion chemistry example for a temperature of 3000 degrees
*****************************************************************************

  * Usage:
        comb3000s(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 16 solutions found in 0.707914 seconds (with a Bezout bound of 96).

Reference: A.P. Morgan. Solving Polynomial Systems Using Continuation for
Engineering and Scientific Problems', Prentice-Hall, Englewood Cliffs, N.J.,
1987. Chapter 9.

See also: http://homepages.math.uic.edu/~jan/Demo/comb3000s.html

+-----------------------------------------------------------------------------+
|i1 : comb3000s(RR_53)                                                        |
|                                                                             |
|                                                                             |
|o1 = {.101815x  + .98961x  + 1.34637x  + 3.4697x   - 2.12454, .57674x  +     |
|              2          6           9          10                   3       |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .78995x  - 2.19493, .087316x  + .895129x  + 1.37722x  + 2.45208x  +     |
|            8                    1           3           5           8       |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .719622x  + .927261x   - 5.67773, .0025003x  + 19.9988x  - 19.9988, -   |
|             9           10                     4           7                |
|     ------------------------------------------------------------------------|
|             2                       2                       2               |
|     1.37722x  + .726099x , - .98961x  + 1.0105x , - 19.9988x  + .050003x , -|
|             1           5           2          6            4           7   |
|     ------------------------------------------------------------------------|
|                                                                     2       |
|     1.93702x x  + .516256x , - .968878x x  + 1.03212x , - 3.21732x x  +     |
|             1 3           8            1 2           9            1 2       |
|     ------------------------------------------------------------------------|
|     .310818x  }                                                             |
|             10                                                              |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use comb3000s:
======================

  * "comb3000s(Ring)"

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

The object comb3000s, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/comb3000s.m2:52:0.

cyclic -- an example of a square polynomial system
**************************************************

  * Usage:
        cyclic(n,kk)
  * Inputs:
      * n, an integer (Macaulay2Doc), the number of variables
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

Cyclic n-roots is a popular benchmark problem. It has finitely many solutions
iff n is square free. In this case the number of solutions is less than the
Bezout bound.

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 70 solutions found in 2.737 seconds with 5 variables.

+-----------------------------------------------------------------------------+
|i1 : cyclic(5,QQ)                                                            |
|                                                                             |
|o1 = {a + b + c + d + e, a*b + b*c + c*d + a*e + d*e, a*b*c + b*c*d + a*b*e +|
|     ------------------------------------------------------------------------|
|     a*d*e + c*d*e, a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e,         |
|     ------------------------------------------------------------------------|
|     a*b*c*d*e - 1}                                                          |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use cyclic:
===================

  * "cyclic(ZZ,Ring)"

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

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

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

The source of this document is in ExampleSystems/cyclic.m2:42:0.

dipole2 -- Dipole2 system
*************************

  * Usage:
        dipole2(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-8250U CPU at 1.60GHz.

There were 4 solutions found in 22.0781 seconds (with a Bezout bound of 576).

Reference: COCONUT, modified by COPRIN.

See also: http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/node1.html#
SECTION000114000000000000000

+-----------------------------------------------------------------------------+
|i1 : dipole2(RR_53)                                                          |
|                                                                             |
|                                                                             |
|o1 = {a + b - .63254, c + d + 1.34534, a*t + b*u - c*v - d*w + .836535, c*t +|
|     ------------------------------------------------------------------------|
|                                   2      2               2               2  |
|     d*u + a*v + b*w - 1.73453, a*t  + b*u  - 2c*t*v - a*v  - 2d*u*w - b*w  -|
|     ------------------------------------------------------------------------|
|                 2      2               2               2               3    |
|     1.35235, c*t  + d*u  + 2a*t*v - c*v  + 2b*u*w - d*w  + .843453, a*t  +  |
|     ------------------------------------------------------------------------|
|        3       2          2      3       2          2      3               3|
|     b*u  - 3c*t v - 3a*t*v  + c*v  - 3d*u w - 3b*u*w  + d*w  + .956345, c*t |
|     ------------------------------------------------------------------------|
|          3       2          2      3       2          2      3              |
|     + d*u  + 3a*t v - 3c*t*v  - a*v  + 3b*u w - 3d*u*w  - b*w  - 1.23425}   |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use dipole2:
====================

  * "dipole2(Ring)"

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

The object dipole2, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/dipole2.m2:47:0.

eco8 -- an 8-dimensional economics problem
******************************************

  * Usage:
        eco8(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 64 solutions found in 9.197 seconds (with a Bezout bound of 1458).

Reference: "Solving polynomial systems using continuation for engineering and
scientific problems" by Alexander Morgan (p 148).

See also: http://homepages.math.uic.edu/~jan/Demo/eco8.html.

+-----------------------------------------------------------------------------+
|i1 : eco8(QQ)                                                                |
|                                                                             |
|o1 = {x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x  - 1, x x x  |
|       1 2 8    2 3 8    3 4 8    4 5 8    5 6 8    6 7 8    1 8       1 3 8 |
|     ------------------------------------------------------------------------|
|     + x x x  + x x x  + x x x  + x x x  + x x  - 2, x x x  + x x x  + x x x |
|        2 4 8    3 5 8    4 6 8    5 7 8    2 8       1 4 8    2 5 8    3 6 8|
|     ------------------------------------------------------------------------|
|     + x x x  + x x  - 3, x x x  + x x x  + x x x  + x x  - 4, x x x  +      |
|        4 7 8    3 8       1 5 8    2 6 8    3 7 8    4 8       1 6 8        |
|     ------------------------------------------------------------------------|
|     x x x  + x x  - 5, x x x  + x x  - 6, x x  - 7, x  + x  + x  + x  + x  +|
|      2 7 8    5 8       1 7 8    6 8       7 8       1    2    3    4    5  |
|     ------------------------------------------------------------------------|
|     x  + x  + 1}                                                            |
|      6    7                                                                 |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use eco8:
=================

  * "eco8(Ring)"

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

The object eco8, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/eco8.m2:47:0.

geneig -- generalized eigenvalue problem
****************************************

  * Usage:
        geneig(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 10 solutions found in 1.447 seconds (with a Bezout bound of 243).

Reference: "Homotopy method for general lambda-matrix problems" by M. Chu, T.Y.
Li and T. Sauer (pages 528-536).

See also: http://homepages.math.uic.edu/~jan/Demo/geneig.html.

+-----------------------------------------------------------------------------+
|i1 : geneig(QQ)                                                              |
|                                                                             |
|             2       2      2      2       2                                 |
|o1 = {- 10x x  + 2x x  - x x  + x x  + 3x x  + x x  + 2x x  + x x  + 2x x  + |
|           1 6     2 6    3 6    4 6     5 6    1 6     2 6    3 6     4 6   |
|     ------------------------------------------------------------------------|
|                                             2        2       2       2      |
|     x x  + 10x  + 2x  - x  + 2x  - 2x , 2x x  - 11x x  + 2x x  - 2x x  +    |
|      5 6      1     2    3     4     5    1 6      2 6     3 6     4 6      |
|     ------------------------------------------------------------------------|
|        2                                                                    |
|     x x  + 2x x  + x x  + 2x x  + x x  + 3x x  + 2x  + 9x  + 3x  - x  - 2x ,|
|      5 6     1 6    2 6     3 6    4 6     5 6     1     2     3    4     5 |
|     ------------------------------------------------------------------------|
|          2       2        2      2      2                                   |
|     - x x  + 2x x  - 12x x  - x x  + x x  + x x  + 2x x  - 2x x  - 2x x  -  |
|        1 6     2 6      3 6    4 6    5 6    1 6     2 6     4 6     5 6    |
|     ------------------------------------------------------------------------|
|                                    2       2      2        2       2        |
|     x  + 3x  + 10x  + 2x  - x , x x  - 2x x  - x x  - 10x x  + 2x x  + 2x x |
|      1     2      3     4    5   1 6     2 6    3 6      4 6     5 6     1 6|
|     ------------------------------------------------------------------------|
|                                                                      2      |
|     + x x  - 2x x  + 2x x  + 3x x  + 2x  - x  + 2x  + 12x  + x , 3x x  +    |
|        2 6     3 6     4 6     5 6     1    2     3      4    5    1 6      |
|     ------------------------------------------------------------------------|
|        2      2       2        2                                            |
|     x x  + x x  + 2x x  - 11x x  + x x  + 3x x  - 2x x  + 3x x  + 3x x  -   |
|      2 6    3 6     4 6      5 6    1 6     2 6     3 6     4 6     5 6     |
|     ------------------------------------------------------------------------|
|     2x  - 2x  - x  + x  + 10x , x  + x  + x  + x  + x  - 1}                 |
|       1     2    3    4      5   1    2    3    4    5                      |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use geneig:
===================

  * "geneig(Ring)"

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

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

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

The source of this document is in ExampleSystems/geneig.m2:49:0.

heart -- an 8-dimensional economics problem
*******************************************

  * Usage:
        heart(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 2 solutions found in 3.356 seconds (with a Bezout bound of 1458).

Reference: "Determination of magnitudes, directions, and locations of two
independent dipoles in a circular conducting region from boundary potential
measurements" by C.V. Nelson and B.C. Hodgkin (pages 817-823),
"Coefficient-Parameter Polynomial Continuation" by A.P. Morgan and A.J. Sommese
(pages 123-160), "Mathematical reduction of a heart dipole model" by A.P.
Morgan and A. Sommese and L.T. Watson (pages 407-410).

See also: http://homepages.math.uic.edu/~jan/Demo/heart.html.

+-----------------------------------------------------------------------------+
|i1 : heart(RR_53)                                                            |
|                                                                             |
|o1 = {a + b - .63254, c + d + 1.34534, a*t + b*u - c*v - d*w + .836535, c*t +|
|     ------------------------------------------------------------------------|
|     d*u + a*v + b*w - 1.73453, | -2ctv-2duw+2at+2bu-2av-2bw-1.35235 |, |    |
|     ------------------------------------------------------------------------|
|     2atv+2buw+2ct+2du-2cv-2dw+.843453 |, |                                  |
|     ------------------------------------------------------------------------|
|     -6atv-6ctv-6buw-6duw+3at+3bu+3cv+3dw+.956345 |, |                       |
|     ------------------------------------------------------------------------|
|     6atv-6ctv+6buw-6duw+3ct+3du-3av-3bw-1.23425 |}                          |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use heart:
==================

  * "heart(Ring)"

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

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

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

The source of this document is in ExampleSystems/heart.m2:48:0.

ipp -- six-revolute-joint problem of mechanics
**********************************************

  * Usage:
        ipp(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 48 solutions found in 1.708 seconds (with a Bezout bound of 256).

Reference: "Computing all solutions to polynomial systems using homotopy
continuation" by A. Morgan and A. Sommese (pages 115-138).

See also: http://homepages.math.uic.edu/~jan/Demo/ipp.html.

+-----------------------------------------------------------------------------+
|i1 : ipp(RR_53)                                                              |
|                                                                             |
|       2    2       2    2       2    2       2    2                         |
|o1 = {x  + x  - 1, x  + x  - 1, x  + x  - 1, x  + x  - 1, - .249151x x  +    |
|       1    2       3    4       5    6       7    8                1 3      |
|     ------------------------------------------------------------------------|
|     .279423x x  + 1.60914x x  + 1.4348x x  - .800528x x  + .400264x x  +    |
|             2 3           1 4          2 4           6 7           5 8      |
|     ------------------------------------------------------------------------|
|     .0740524x  - .08305x  - .38616x  - .755266x  + .504202x  - 1.09163x  +  |
|              1          2          3           4           5           6    |
|     ------------------------------------------------------------------------|
|     .400264x  + .0492073, .125016x x  - .119228x x  - .686607x x  -         |
|             8                     1 3           2 3           1 4           |
|     ------------------------------------------------------------------------|
|     .71994x x  - .432419x x  - .864839x x  - .0371573x  + .0354369x  +      |
|            2 4           5 7           6 8            1            2        |
|     ------------------------------------------------------------------------|
|     .0853835x  - .039252x  - .432419x  + .013873, - .63555x x  - .666404x x |
|              3           5           7                     1 3           2 3|
|     ------------------------------------------------------------------------|
|     - .11572x x  + .110362x x  + .290702x x  - .629388x x  + 1.25878x x  +  |
|              1 4           2 4           5 7           6 7           5 8    |
|     ------------------------------------------------------------------------|
|     .581404x x  + .195947x  - 1.22803x  - .0790342x  + .0263879x  -         |
|             6 8           1           2            4            5           |
|     ------------------------------------------------------------------------|
|     .0571314x  - 1.16281x  + 1.25878x  + 2.16257, 1.48948x x  + 1.32811x x  |
|              6           7           8                    1 3           2 3 |
|     ------------------------------------------------------------------------|
|     + .230623x x  - .258645x x  + 1.16517x x  + .53817x x  - .269085x x  +  |
|               1 4           2 4           5 7          6 7           5 8    |
|     ------------------------------------------------------------------------|
|     .582586x x  - .20817x  + 2.68683x  - .699103x  + .357444x  + 1.24991x  +|
|             6 8          1           2           3           4           5  |
|     ------------------------------------------------------------------------|
|     1.46774x  + 1.16517x  + 1.10763x  - .696868}                            |
|             6           7           8                                       |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use ipp:
================

  * "ipp(Ring)"

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

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

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

The source of this document is in ExampleSystems/ipp.m2:48:0.

katsura -- an example of a 0-dimensional square polynomial system
*****************************************************************

  * Usage:
        katsura(n,kk)
  * Inputs:
      * n, an integer (Macaulay2Doc), the number of variables
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 512 solutions found in 2.804 seconds with 10 variables.

The unknowns stand for the values (in [0,1]) of a distribution function of a
field created by a mixture of a ferro-antiferro-magnetic bond at some points.

+-----------------------------------------------------------------------------+
|i1 : katsura(10,QQ)                                                          |
|                                                                             |
|                                                           2     2     2     |
|o1 = {a + 2b + 2c + 2d + 2e + 2f + 2g + 2h + 2i + 2j - 1, a  + 2b  + 2c  +   |
|     ------------------------------------------------------------------------|
|       2     2     2     2     2     2     2                                 |
|     2d  + 2e  + 2f  + 2g  + 2h  + 2i  + 2j  - a, 2a*b + 2b*c + 2c*d + 2d*e +|
|     ------------------------------------------------------------------------|
|                                            2                                |
|     2e*f + 2f*g + 2g*h + 2h*i + 2i*j - b, b  + 2a*c + 2b*d + 2c*e + 2d*f +  |
|     ------------------------------------------------------------------------|
|                                                                             |
|     2e*g + 2f*h + 2g*i + 2h*j - c, 2b*c + 2a*d + 2b*e + 2c*f + 2d*g + 2e*h +|
|     ------------------------------------------------------------------------|
|                       2                                                     |
|     2f*i + 2g*j - d, c  + 2b*d + 2a*e + 2b*f + 2c*g + 2d*h + 2e*i + 2f*j -  |
|     ------------------------------------------------------------------------|
|                                                             2               |
|     e, 2c*d + 2b*e + 2a*f + 2b*g + 2c*h + 2d*i + 2e*j - f, d  + 2c*e + 2b*f |
|     ------------------------------------------------------------------------|
|                                                                             |
|     + 2a*g + 2b*h + 2c*i + 2d*j - g, 2d*e + 2c*f + 2b*g + 2a*h + 2b*i + 2c*j|
|     ------------------------------------------------------------------------|
|           2                                                                 |
|     - h, e  + 2d*f + 2c*g + 2b*h + 2a*i + 2b*j - i}                         |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use katsura:
====================

  * "katsura(ZZ,Ring)"

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

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

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

The source of this document is in ExampleSystems/katsura.m2:49:0.

ku10 -- a 10-dimensional system of Ku
*************************************

  * Usage:
        ku10(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 4 solutions found in 2.266 seconds (with a Bezout bound of 1024).

We point out that this system is difficult for homotopy continuation (without
multi-homogenization), yet an easy system for elimination.

Reference: "Die numeriese oplos van stelsels polinoomvergelykings" by M.C.
Steenkamp.

See also: http://homepages.math.uic.edu/~jan/Demo/ku10.html.

+-----------------------------------------------------------------------------+
|i1 : ku10(QQ)                                                                |
|                                                                             |
|o1 = {5x x  + 5x  + 3x  + 55, 7x x  + 9x  + 9x  + 19, 3x x  + 6x  + 5x  - 4, |
|        1 2     1     2         2 3     2     3         3 4     3     4      |
|     ------------------------------------------------------------------------|
|     6x x  + 6x  + 7x  + 118, x x  + 3x  + 9x  + 27, 6x x  + 7x  + x  + 72,  |
|       4 5     4     5         5 6     5     6         6 7     6    7        |
|     ------------------------------------------------------------------------|
|     9x x  + 7x  + x  + 35, 4x x  + 4x  + 6x  + 16, 8x x   + 4x  + 3x   - 51,|
|       7 8     7    8         8 9     8     9         9 10     9     10      |
|     ------------------------------------------------------------------------|
|     3x x   - 6x  + x   + 5}                                                 |
|       1 10     1    10                                                      |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use ku10:
=================

  * "ku10(Ring)"

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

The object ku10, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/ku10.m2:51:0.

lorentz -- equilibrium points of a 4-dimensional Lorentz attractor
******************************************************************

  * Usage:
        lorentz(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 12 solutions found in 0.0562 seconds (with a Bezout bound of 16).

Reference: "Solving polynomial systems" by Tien-Yien Li (pages 33-39).

See also: http://homepages.math.uic.edu/~jan/Demo/lorentz.html.

+-----------------------------------------------------------------------------+
|i1 : lorentz(QQ)                                                             |
|                                                                             |
|o1 = {x x  - x x  - x  + 1, x x  - x x  - x  + 1, - x x  + x x  - x  + 1,    |
|       1 2    1 3    4       2 3    2 4    1         1 3    3 4    2         |
|     ------------------------------------------------------------------------|
|     x x  - x x  - x  + 1}                                                   |
|      1 4    2 4    3                                                        |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use lorentz:
====================

  * "lorentz(Ring)"

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

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

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

The source of this document is in ExampleSystems/lorentz.m2:42:0.

lumped -- lumped parameter chemically reacting system
*****************************************************

  * Usage:
        lumped(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 4 solutions found in 0.178 seconds (with a Bezout bound of 16).

Reference: "Solving deficient polynomial systems with homotopies which keep the
subschemes at infinity invariant" by T.Y. Li and X. Wang (pages 693-710).

See also: http://homepages.math.uic.edu/~jan/Demo/lumped.html.

+-----------------------------------------------------------------------------+
|i1 : lumped(CC_53)                                                           |
|                                                                             |
|o1 = {(.767719 + .328203*ii)x x  + (.767719 + .328203*ii)x x  + (- 1.76772 - |
|                             1 3                          1 4                |
|     ------------------------------------------------------------------------|
|     .328203*ii)x  + (.548909 + .109395*ii)x  + .0479608 + .888687*ii, (.3301|
|                 1                          3                                |
|     ------------------------------------------------------------------------|
|     + .890584*ii)x x  + (.3301 + .890584*ii)x x  + (- 1.3301 - .890584*ii)x |
|                   2 3                        2 4                           2|
|     ------------------------------------------------------------------------|
|     + (.111291 + .671775*ii)x  + .829152 + .669877*ii, (- .767719 -         |
|                              4                                              |
|     ------------------------------------------------------------------------|
|     .328203*ii)x x  + (- .767719 - .328203*ii)x x  + (- .892482 -           |
|                 1 3                            1 4                          |
|     ------------------------------------------------------------------------|
|     .452966*ii)x x  + (.767719 + .328203*ii)x  + (- .548909 - .109395*ii)x ,|
|                 3 4                          1                            3 |
|     ------------------------------------------------------------------------|
|     (- .3301 - .890584*ii)x x  + (- .3301 - .890584*ii)x x  + (- .892482 -  |
|                            2 3                          2 4                 |
|     ------------------------------------------------------------------------|
|     .452966*ii)x x  + (.3301 + .890584*ii)x  + (- .111291 - .671775*ii)x }  |
|                 3 4                        2                            4   |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use lumped:
===================

  * "lumped(Ring)"

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

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

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

The source of this document is in ExampleSystems/lumped.m2:43:0.

noon5 -- A neural network modeled by an adaptive Lotka-Volterra system with 5 variables
***************************************************************************************

  * Usage:
        noon5(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 233 solutions found in 0.582 seconds (with a Bezout bound of 243).

The coefficients have been chosen so that full permutation symmetry was
obtained.

Reference: "A neural network modeled by an adaptive Lotka-Volterra system" by
V.W. Noonburg.

See also: http://homepages.math.uic.edu/~jan/Demo/noon5.html.

+-----------------------------------------------------------------------------+
|i1 : noon5(RR_53)                                                            |
|                                                                             |
|         2      2      2      2               2        2      2      2       |
|o1 = {x x  + x x  + x x  + x x  - 1.1x  + 1, x x  + x x  + x x  + x x  -     |
|       1 2    1 3    1 4    1 5       1       1 2    2 3    2 4    2 5       |
|     ------------------------------------------------------------------------|
|                 2      2        2      2               2      2      2      |
|     1.1x  + 1, x x  + x x  + x x  + x x  - 1.1x  + 1, x x  + x x  + x x  +  |
|         2       1 3    2 3    3 4    3 5       3       1 4    2 4    3 4    |
|     ------------------------------------------------------------------------|
|        2               2      2      2      2                               |
|     x x  - 1.1x  + 1, x x  + x x  + x x  + x x  - 1.1x  + 1}                |
|      4 5       4       1 5    2 5    3 5    4 5       5                     |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use noon5:
==================

  * "noon5(Ring)"

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

The object noon5, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/noon5.m2:44:0.

proddeco -- system with a product-decomposition structure
*********************************************************

  * Usage:
        prodecco(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 68 solutions found in 7.236 seconds (with a Bezout bound of 256).

Reference: "A product-decomposition theorem for bounding Bezout numbers" by
A.P. Morgan, A.J. Sommese, and C.W. Wampler (pages 1308-1325).

See also: http://homepages.math.uic.edu/~jan/Demo/proddeco.html.

+-----------------------------------------------------------------------------+
|i1 : proddeco(RR_53)                                                         |
|                                                                             |
|              2 2                             2 2                            |
|o1 = {64.6329z z  - 129.266z z z z  + 64.6329z z  - 55.6619z z z  -          |
|              2 3           1 2 3 4           1 4           1 2 3            |
|     ------------------------------------------------------------------------|
|            2               2           2                                    |
|     12.201z z  - 48.2115z z  + 55.6619z z  + 12.201z z z  + 48.2115z z z  - |
|            2 3           2 3           1 4          1 2 4           1 3 4   |
|     ------------------------------------------------------------------------|
|                             2                                               |
|     71.399z z z  + 71.399z z  + 50.7966z z  + 9.14137z z  + 62.3536z z  +   |
|            2 3 4          1 4           1 3           2 3           1 4     |
|     ------------------------------------------------------------------------|
|                          2 2                             2 2                |
|     17.5403z z , 81.3704z z  - 162.741z z z z  + 81.3704z z  - 55.3638z z z |
|             2 4          2 3           1 2 3 4           1 4           1 2 3|
|     ------------------------------------------------------------------------|
|               2               2           2                                 |
|     - 25.0326z z  - 21.7819z z  + 55.3638z z  + 25.0326z z z  +             |
|               2 3           2 3           1 4           1 2 4               |
|     ------------------------------------------------------------------------|
|                                               2                             |
|     21.7819z z z  - 55.0737z z z  + 55.0737z z  + 24.7967z z  + 9.562z z  + |
|             1 3 4           2 3 4           1 4           1 3         2 3   |
|     ------------------------------------------------------------------------|
|                                        2 2                             2 2  |
|     41.6882z z  + 20.9301z z , 35.2817z z  - 70.5634z z z z  + 35.2817z z  -|
|             1 4           2 4          2 3           1 2 3 4           1 4  |
|     ------------------------------------------------------------------------|
|                             2               2           2                   |
|     42.2584z z z  - 23.8847z z  - 8.07978z z  + 42.2584z z  + 23.8847z z z  |
|             1 2 3           2 3           2 3           1 4           1 2 4 |
|     ------------------------------------------------------------------------|
|                                                 2                           |
|     + 8.07978z z z  - 23.1781z z z  + 23.1781z z  + 13.503z z  + 10.3468z z |
|               1 3 4           2 3 4           1 4          1 3           2 3|
|     ------------------------------------------------------------------------|
|                                          2 2                             2 2|
|     + 37.0338z z  + 16.9465z z , 6.89936z z  - 13.7987z z z z  + 6.89936z z |
|               1 4           2 4          2 3           1 2 3 4           1 4|
|     ------------------------------------------------------------------------|
|                               2               2           2                 |
|     - 43.1035z z z  - 56.0575z z  - 2.37993z z  + 43.1035z z  +             |
|               1 2 3           2 3           2 3           1 4               |
|     ------------------------------------------------------------------------|
|                                                               2             |
|     56.0575z z z  + 2.37993z z z  - 6.40051z z z  + 6.40051z z  +           |
|             1 2 4           1 3 4           2 3 4           1 4             |
|     ------------------------------------------------------------------------|
|     16.2879z z  + 22.5597z z  + 42.9769z z  + 55.4137z z }                  |
|             1 3           2 3           1 4           2 4                   |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use proddeco:
=====================

  * "proddeco(Ring)"

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

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

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

The source of this document is in ExampleSystems/proddeco.m2:41:0.

puma -- hand position and orientation of PUMA robot
***************************************************

  * Usage:
        puma(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 16 solutions found in 0.531 seconds (with a Bezout bound of 128).

Reference: "Box-Bisection for solving second-degree systems and the problem of
clustering', ACM Transactions on Mathematical Software" by A. Morgan and V.
Shapiro. (p. 152-167).

See also: http://homepages.math.uic.edu/~jan/Demo/puma.html.

+-----------------------------------------------------------------------------+
|i1 : puma(RR_53)                                                             |
|                                                                             |
|       2    2       2    2       2    2       2    2                         |
|o1 = {x  + x  - 1, x  + x  - 1, x  + x  - 1, x  + x  - 1, .004731x x  -      |
|       1    2       3    4       5    6       7    8              1 3        |
|     ------------------------------------------------------------------------|
|     .3578x x  - .1238x  - .001637x  - .9338x  + x  - .3571, .2238x x  +     |
|           2 3         1           2         4    7                1 3       |
|     ------------------------------------------------------------------------|
|     .7623x x  + .2638x  - .07745x  - .6734x  - .6022, x x  + .3578x  +      |
|           2 3         1          2         4           6 8         1        |
|     ------------------------------------------------------------------------|
|     .004731x , - .7623x  + .2238x  + .3461}                                 |
|             2          1         2                                          |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use puma:
=================

  * "puma(Ring)"

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

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

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

The source of this document is in ExampleSystems/puma.m2:45:0.

quadgrid -- interpolating quadrature formula for function defined on a grid
***************************************************************************

  * Usage:
        quadgrid(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 5 solutions found in 1.653 seconds (with a Bezout bound of 120).

Note: This system is ill-conditioned. There are 4 complex and 1 real solution.

Reference: "The construction and application of wavelets in numerical analysis"
by Wim Sweldens.

See also: http://homepages.math.uic.edu/~jan/Demo/quadgrid.html.

+-----------------------------------------------------------------------------+
|i1 : quadgrid(RR_53)                                                         |
|                                                                             |
|                                                                             |
|o1 = {w  + w  + w  + w  - 1, w b + w b + w b + w b + .5w  + w  + 1.5w  -     |
|       0    1    2    3       0     1     2     3       1    2       3       |
|     ------------------------------------------------------------------------|
|                 2      2      2      2                                      |
|     .633975, w b  + w b  + w b  + w b  + w b + 2w b + 3w b + .25w  + w  +   |
|               0      1      2      3      1      2      3        1    2     |
|     ------------------------------------------------------------------------|
|                         3      3      3      3         2       2         2  |
|     4.5w  - .401924, w b  + w b  + w b  + w b  + 1.5w b  + 3w b  + 4.5w b  +|
|         3             0      1      2      3         1       2         3    |
|     ------------------------------------------------------------------------|
|                                                                   4      4  |
|     .75w b + 3w b + 6.75w b + .125w  + w  + 3.375w  - .131092, w b  + w b  +|
|         1      2         3         1    2         3             0      1    |
|     ------------------------------------------------------------------------|
|        4      4       3       3       3         2       2          2        |
|     w b  + w b  + 2w b  + 4w b  + 6w b  + 1.5w b  + 6w b  + 13.5w b  + .5w b|
|      2      3       1       2       3         1       2          3        1 |
|     ------------------------------------------------------------------------|
|     + 4w b + 13.5w b + .0625w  + w  + 5.0625w  + .302193}                   |
|         2         3          1    2          3                              |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use quadgrid:
=====================

  * "quadgrid(Ring)"

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

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

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

The source of this document is in ExampleSystems/quadgrid.m2:44:0.

rabmo -- optimal multi-dimensional quadrature formulas
******************************************************

  * Usage:
        rabmo(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 58 solutions found in 1642.07 seconds (with a Bezout bound of
36000).

Reference: "Methods and applications of interval analysis" by Ramon E. Moore
(p. 64).

See also: http://homepages.math.uic.edu/~jan/Demo/rabmo.html.

+-----------------------------------------------------------------------------+
|i1 : rabmo(QQ)                                                               |
|                                                                             |
|                                                                    2     2  |
|o1 = {x  + x  + x  + 2x  - 1, x x  + x x  + 2x x  + 2x x  + 2x x  - -, x x  +|
|       1    3    5     7       1 2    3 4     5 6     7 8     7 9   3   1 2  |
|     ------------------------------------------------------------------------|
|        2       2       2       2   2     3      3       3       3       3   |
|     x x  + 2x x  + 2x x  + 2x x  - -, x x  + x x  + 2x x  + 2x x  + 2x x  - |
|      3 4     5 6     7 8     7 9   5   1 2    3 4     5 6     7 8     7 9   |
|     ------------------------------------------------------------------------|
|     2     4      4       4       4       4   2     2             1     4    |
|     -, x x  + x x  + 2x x  + 2x x  + 2x x  - -, x x  + 2x x x  - -, x x  +  |
|     7   1 2    3 4     5 6     7 8     7 9   9   5 6     7 8 9   9   5 6    |
|     ------------------------------------------------------------------------|
|         2 2    1     3      2          2    1     4      3          3    1  |
|     2x x x  - --, x x  + x x x  + x x x  - --, x x  + x x x  + x x x  - --} |
|       7 8 9   25   5 6    7 8 9    7 8 9   15   5 6    7 8 9    7 8 9   21  |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use rabmo:
==================

  * "rabmo(Ring)"

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

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

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

The source of this document is in ExampleSystems/rabmo.m2:47:0.

randomGeneralizedEigenvalueProblem -- an example of a 0-dimensional square polynomial system
********************************************************************************************

  * Usage:
        (T,S,solsS) = randomGeneralizedEigenvalueProblem(n,K)
  * Inputs:
      * n, an integer (Macaulay2Doc), positive value representing the
        number of polynomials
  * Outputs:
      * T'S'solsS, a sequence (Macaulay2Doc), target system T,
        start system S, and solutions solsS to the start system

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

Given $n\times n$ matrices $A$ and $B$, a number $\lambda$ is a generalized
eigenvalue if there is a nonzero vector $v$ such that $A x = \lambda B x$.

This function creates a square target system representing the problem for
random $A$ and $B$ and a start system representing the problem with eigenvalues
that are $n$-th roots of unity and the corresponding eigenvectors form the
standard basis.

This system was solved in May 2020, using track (NumericalAlgebraicGeometry),
in Macaulay2 v1.15
with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

For a system of 3 polynomials there were 3 solutions found in 0.0764 seconds.

For a system of 5 polynomials there were 5 solutions found in 0.0478 seconds.

+-----------------------------------------------------------------------------+
|i1 : randomGeneralizedEigenvalueProblem 3                                    |
|                                                                             |
|o1 = ({(- .741046 - .108386*ii)lambda*x1 + (- .830833 - .538554*ii)lambda*x2 |
|     ------------------------------------------------------------------------|
|     + (- .191734 - .403215*ii)lambda*x3 + (.892712 + .673395*ii)x1 + (.89189|
|     ------------------------------------------------------------------------|
|     + .231053*ii)x2 + (.0741835 + .808694*ii)x3, (- .348931 -               |
|     ------------------------------------------------------------------------|
|     .562428*ii)lambda*x1 + (- .873665 - .415912*ii)lambda*x2 + (- .615911 - |
|     ------------------------------------------------------------------------|
|     .0147867*ii)lambda*x3 + (.29398 + .632944*ii)x1 + (.461944 +            |
|     ------------------------------------------------------------------------|
|     .775187*ii)x2 + (.362835 + .706096*ii)x3, (- .246268 -                  |
|     ------------------------------------------------------------------------|
|     .153346*ii)lambda*x1 + (- .606588 - .848005*ii)lambda*x2 + (- .223028 - |
|     ------------------------------------------------------------------------|
|     .388829*ii)lambda*x3 + (.0258884 + .714827*ii)x1 + (.909047 +           |
|     ------------------------------------------------------------------------|
|     .314897*ii)x2 + (.127435 + .254482*ii)x3, - x1 - x2 - x3 + 2},          |
|     ------------------------------------------------------------------------|
|     {lambda*x1 - lambda - x1 + 1, lambda*x2 - lambda + (.5 - .866025*ii)x2 -|
|     ------------------------------------------------------------------------|
|     .5 + .866025*ii, lambda*x3 - lambda + (.5 + .866025*ii)x3 - .5 -        |
|     ------------------------------------------------------------------------|
|     .866025*ii, - x1 - x2 - x3 + 2}, {{1, 0, 1, 1}, {-.5+.866025*ii, 1, 0,  |
|     ------------------------------------------------------------------------|
|     1}, {-.5-.866025*ii, 1, 1, 0}})                                         |
|                                                                             |
|o1 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i2 : randomGeneralizedEigenvalueProblem 5                                    |
|                                                                             |
|o2 = ({(- .977573 - .212436*ii)lambda*x1 + (- .205375 - .276652*ii)lambda*x2 |
|     ------------------------------------------------------------------------|
|     + (- .130004 - .522285*ii)lambda*x3 + (- .971588 - .249992*ii)lambda*x4 |
|     ------------------------------------------------------------------------|
|     + (- .581271 - .640172*ii)lambda*x5 + (.557119 + .873708*ii)x1 +        |
|     ------------------------------------------------------------------------|
|     (.169813 + .965004*ii)x2 + (.350611 + .379495*ii)x3 + (.184779 +        |
|     ------------------------------------------------------------------------|
|     .370833*ii)x4 + (.305423 + .732358*ii)x5, (- .592747 -                  |
|     ------------------------------------------------------------------------|
|     .831802*ii)lambda*x1 + (- .0958269 - .605398*ii)lambda*x2 + (- .171029 -|
|     ------------------------------------------------------------------------|
|     .340019*ii)lambda*x3 + (- .0328338 - .21113*ii)lambda*x4 + (- .919548 - |
|     ------------------------------------------------------------------------|
|     .560684*ii)lambda*x5 + (.7037 + .681869*ii)x1 + (.0647412 +             |
|     ------------------------------------------------------------------------|
|     .877846*ii)x2 + (.237252 + .116721*ii)x3 + (.339222 + .062212*ii)x4 +   |
|     ------------------------------------------------------------------------|
|     (.562839 + .629991*ii)x5, (- .501243 - .154289*ii)lambda*x1 + (- .883549|
|     ------------------------------------------------------------------------|
|     - .942865*ii)lambda*x2 + (- .818142 - .781168*ii)lambda*x3 + (- .611415 |
|     ------------------------------------------------------------------------|
|     - .638389*ii)lambda*x4 + (- .847715 - .763255*ii)lambda*x5 + (.276259 + |
|     ------------------------------------------------------------------------|
|     .605659*ii)x1 + (.0340514 + .507989*ii)x2 + (.444183 + .644366*ii)x3 +  |
|     ------------------------------------------------------------------------|
|     (.465736 + .40273*ii)x4 + (.479826 + .815167*ii)x5, (- .467203 -        |
|     ------------------------------------------------------------------------|
|     .765564*ii)lambda*x1 + (- .0821679 - .10394*ii)lambda*x2 + (- .767433 - |
|     ------------------------------------------------------------------------|
|     .155346*ii)lambda*x3 + (- .0594513 - .0625324*ii)lambda*x4 + (- .477291 |
|     ------------------------------------------------------------------------|
|     - .0048212*ii)lambda*x5 + (.96518 + .681683*ii)x1 + (.150294 +          |
|     ------------------------------------------------------------------------|
|     .656391*ii)x2 + (.194945 + .518585*ii)x3 + (.164647 + .713493*ii)x4 +   |
|     ------------------------------------------------------------------------|
|     (.97723 + .0595849*ii)x5, (- .305946 - .53632*ii)lambda*x1 + (- .280679 |
|     ------------------------------------------------------------------------|
|     - .475179*ii)lambda*x2 + (- .0215389 - .283851*ii)lambda*x3 + (- .270725|
|     ------------------------------------------------------------------------|
|     - .075503*ii)lambda*x4 + (- .46452 - .16645*ii)lambda*x5 + (.914199 +   |
|     ------------------------------------------------------------------------|
|     .887381*ii)x1 + (.174853 + .626892*ii)x2 + (.987173 + .568273*ii)x3 +   |
|     ------------------------------------------------------------------------|
|     (.909537 + .566034*ii)x4 + (.0645275 + .283709*ii)x5, - x1 - x2 - x3 -  |
|     ------------------------------------------------------------------------|
|     x4 - x5 + 4}, {lambda*x1 - lambda - x1 + 1, lambda*x2 - lambda + (-     |
|     ------------------------------------------------------------------------|
|     .309017 - .951057*ii)x2 + .309017 + .951057*ii, lambda*x3 - lambda +    |
|     ------------------------------------------------------------------------|
|     (.809017 - .587785*ii)x3 - .809017 + .587785*ii, lambda*x4 - lambda +   |
|     ------------------------------------------------------------------------|
|     (.809017 + .587785*ii)x4 - .809017 - .587785*ii, lambda*x5 - lambda + (-|
|     ------------------------------------------------------------------------|
|     .309017 + .951057*ii)x5 + .309017 - .951057*ii, - x1 - x2 - x3 - x4 - x5|
|     ------------------------------------------------------------------------|
|     + 4}, {{1, 0, 1, 1, 1, 1}, {.309017+.951057*ii, 1, 0, 1, 1, 1},         |
|     ------------------------------------------------------------------------|
|     {-.809017+.587785*ii, 1, 1, 0, 1, 1}, {-.809017-.587785*ii, 1, 1, 1, 0, |
|     ------------------------------------------------------------------------|
|     1}, {.309017-.951057*ii, 1, 1, 1, 1, 0}})                               |
|                                                                             |
|o2 : Sequence                                                                |
+-----------------------------------------------------------------------------+

Ways to use randomGeneralizedEigenvalueProblem:
===============================================

  * "randomGeneralizedEigenvalueProblem(ZZ)"

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

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

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

The source of this document is in
ExampleSystems/randomGeneralizedEigenvalueProblem.m2:54:0.

randomSystem -- an example of a 0-dimensional square polynomial system
**********************************************************************

  * Usage:
        F = randomSystem(n,d,K)
  * Inputs:
      * n, an integer (Macaulay2Doc), positive value representing the
        number of polynomials
      * d, an integer (Macaulay2Doc), positive value representing the
        degree of the polynomials
      * K, a ring (Macaulay2Doc), usually a field, e.g., QQ or
        CC_{53}
  * Outputs:
      * F, a list (Macaulay2Doc), a system of $n$ polynomials
        random polynomials of degree $d$ in $n$ variables.

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

This system was solved in May 2020, using track (NumericalAlgebraicGeometry),
in Macaulay2 v1.15
with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

For a system with 2 polynomials of degree 3 over rationals there were 9
solutions found in 0.06 seconds.

For a system with 5 polynomials of degree 2 over complex numbers there were 32
solutions found in 0.297 seconds.

+-----------------------------------------------------------------------------+
|i1 : randomSystem(2,3,QQ)                                                    |
|                                                                             |
|       7  3    7  2     7     2      3   3  2   7        7  2   9     9      |
|o1 = {--x1  + --x1 x2 + -x1*x2  + 7x2  + -x1  + -x1*x2 + -x2  + -x1 + -x2 -  |
|      10      10        3                4      4        9      2     4      |
|     ------------------------------------------------------------------------|
|         3  3   3  2           2   10  3      2   5        2  2   3     6    |
|     1, --x1  + -x1 x2 + 5x1*x2  + --x2  + 6x1  + -x1*x2 + -x2  + -x1 + -x2 -|
|        10      7                   9             4        9      7     7    |
|     ------------------------------------------------------------------------|
|     1}                                                                      |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+
|i2 : randomSystem(5, 2, CC_53)                                               |
|-- warning: experimental computation over inexact field begun                |
|--          results not reliable (one warning given per session)             |
|                                                                             |
|                              2                                              |
|o2 = {(.174853 + .626892*ii)x1  + (.350611 + .379495*ii)x1*x2 + (.987173 +   |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .568273*ii)x2  + (.237252 + .116721*ii)x1*x3 + (.184779 +               |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .370833*ii)x2*x3 + (.164647 + .713493*ii)x3  + (.444183 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .644366*ii)x1*x4 + (.339222 + .062212*ii)x2*x4 + (.909537 +             |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .566034*ii)x3*x4 + (.562839 + .629991*ii)x4  + (.194945 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .518585*ii)x1*x5 + (.465736 + .40273*ii)x2*x5 + (.305423 +              |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .732358*ii)x3*x5 + (.479826 + .815167*ii)x4*x5 + (.97723 +              |
|     ------------------------------------------------------------------------|
|                   2                                                         |
|     .0595849*ii)x5  + (.914199 + .887381*ii)x1 + (.169813 + .965004*ii)x2 + |
|     ------------------------------------------------------------------------|
|                                                                             |
|     (.0647412 + .877846*ii)x3 + (.0340514 + .507989*ii)x4 + (.150294 +      |
|     ------------------------------------------------------------------------|
|                                               2                             |
|     .656391*ii)x5 - 1, (.305946 + .53632*ii)x1  + (.205375 +                |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .276652*ii)x1*x2 + (.280679 + .475179*ii)x2  + (.0958269 +              |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .605398*ii)x1*x3 + (.130004 + .522285*ii)x2*x3 + (.767433 +             |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .155346*ii)x3  + (.883549 + .942865*ii)x1*x4 + (.171029 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .340019*ii)x2*x4 + (.0215389 + .283851*ii)x3*x4 + (.0328338 +           |
|     ------------------------------------------------------------------------|
|                 2                                                           |
|     .21113*ii)x4  + (.0821679 + .10394*ii)x1*x5 + (.818142 +                |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .781168*ii)x2*x5 + (.971588 + .249992*ii)x3*x5 + (.611415 +             |
|     ------------------------------------------------------------------------|
|                                                  2                          |
|     .638389*ii)x4*x5 + (.0594513 + .0625324*ii)x5  + (.0645275 +            |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .283709*ii)x1 + (.977573 + .212436*ii)x2 + (.592747 + .831802*ii)x3 +   |
|     ------------------------------------------------------------------------|
|                                                                             |
|     (.501243 + .154289*ii)x4 + (.467203 + .765564*ii)x5 - 1, (.46452 +      |
|     ------------------------------------------------------------------------|
|                 2                                                           |
|     .16645*ii)x1  + (.224737 + .00415468*ii)x1*x2 + (.905002 +              |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .502056*ii)x2  + (.370963 + .511938*ii)x1*x3 + (.888883 +               |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .0776828*ii)x2*x3 + (.35911 + .751187*ii)x3  + (.806831 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .532303*ii)x1*x4 + (.41855 + .138169*ii)x2*x4 + (.252537 +              |
|     ------------------------------------------------------------------------|
|                                               2                             |
|     .10246*ii)x3*x4 + (.475683 + .107149*ii)x4  + (.572433 +                |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .269298*ii)x1*x5 + (.543508 + .212989*ii)x2*x5 + (.543294 +             |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .659792*ii)x3*x5 + (.13359 + .420228*ii)x4*x5 + (.112527 +              |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .659052*ii)x5  + (.270725 + .075503*ii)x1 + (.581271 + .640172*ii)x2 +  |
|     ------------------------------------------------------------------------|
|                                                                             |
|     (.919548 + .560684*ii)x3 + (.847715 + .763255*ii)x4 + (.477291 +        |
|     ------------------------------------------------------------------------|
|                                                 2                           |
|     .0048212*ii)x5 - 1, (.689783 + .231832*ii)x1  + (.0311856 +             |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .388518*ii)x1*x2 + (.762468 + .455803*ii)x2  + (.0258757 +              |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .0288963*ii)x1*x3 + (.299391 + .77016*ii)x2*x3 + (.429347 +             |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .288364*ii)x3  + (.912808 + .898905*ii)x1*x4 + (.885699 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .761573*ii)x2*x4 + (.993594 + .895324*ii)x3*x4 + (.0770766 +            |
|     ------------------------------------------------------------------------|
|                 2                                                           |
|     .30813*ii)x4  + (.580909 + .228196*ii)x1*x5 + (.42526 + .403967*ii)x2*x5|
|     ------------------------------------------------------------------------|
|                                                                             |
|     + (.34672 + .363895*ii)x3*x5 + (.633138 + .262856*ii)x4*x5 + (.559948 + |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .249212*ii)x5  + (.649378 + .564001*ii)x1 + (.455949 + .00471457*ii)x2 +|
|     ------------------------------------------------------------------------|
|                                                                             |
|     (.771746 + .74002*ii)x3 + (.20217 + .774737*ii)x4 + (.556033 +          |
|     ------------------------------------------------------------------------|
|                                               2                             |
|     .91523*ii)x5 - 1, (.531857 + .356922*ii)x1  + (.172651 +                |
|     ------------------------------------------------------------------------|
|                                                 2                           |
|     .533417*ii)x1*x2 + (.738044 + .0215483*ii)x2  + (.0219272 +             |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .817573*ii)x1*x3 + (.751374 + .767024*ii)x2*x3 + (.886882 +             |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .784812*ii)x3  + (.168645 + .358533*ii)x1*x4 + (.553621 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .74003*ii)x2*x4 + (.829966 + .270696*ii)x3*x4 + (.831694 +              |
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     .729951*ii)x4  + (.255456 + .466765*ii)x1*x5 + (.545222 +               |
|     ------------------------------------------------------------------------|
|                                                                             |
|     .958457*ii)x2*x5 + (.790306 + .783351*ii)x3*x5 + (.29683 +              |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     .162247*ii)x4*x5 + (.908964 + .231486*ii)x5  + (.671836 + .707257*ii)x1 |
|     ------------------------------------------------------------------------|
|     + (.0423322 + .839258*ii)x2 + (.821811 + .68646*ii)x3 + (.101507 +      |
|     ------------------------------------------------------------------------|
|     .253324*ii)x4 + (.223416 + .0263964*ii)x5 - 1}                          |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use randomSystem:
=========================

  * "randomSystem(ZZ,ZZ,Ring)"

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

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

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

The source of this document is in ExampleSystems/randomSystem.m2:41:0.

rbpl24s -- parallel robot with 24 real solutions, after variable scaling
************************************************************************

  * Usage:
        rbpl24s(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 40 solutions found in 3.740 seconds (with a Bezout bound of 576).

Reference: generated by Fabrice Rouillier, as a deformation of a system found
in Innocenti.

See also: http://homepages.math.uic.edu/~jan/Demo/rbpl24s.html.

+-----------------------------------------------------------------------------+
|i1 : rbpl24s(RR_53)                                                          |
|                                                                             |
|              2           2          2                    2           2      |
|o1 = {.902353x  + .506753y  + 1.1541z  - 1.89489, .547653x  + .401593y  +    |
|              1           1          1                    2           2      |
|     ------------------------------------------------------------------------|
|             2                              2           2           2        |
|     .914601z  - 1.374x  - 3.61818, 1.26481x  + .299954y  + 1.20376z  +      |
|             2         2                    3           3           3        |
|     ------------------------------------------------------------------------|
|     .346368x  - 3.03447y  - 2.08335, 1.01543x x  + .651627y y  + 1.48404z z |
|             3           3                    1 2           1 2           1 2|
|     ------------------------------------------------------------------------|
|     - 1.3475x  + .75575, 1.4913x x  + .544239y y  + 1.64533z z  + .1979x  - |
|              2                  1 3           1 3           1 3         3   |
|     ------------------------------------------------------------------------|
|     1.73377y  - 2.1825, 3.63357x x  + 1.51527y y  + 4.58093z z  - 4.97527x  |
|             3                   2 3           2 3           2 3           2 |
|     ------------------------------------------------------------------------|
|     + .45581x  - 3.99328y  - .00437819, 1.74298x y z  - .991213x y z  -     |
|              3           3                      3 2 1           2 3 1       |
|     ------------------------------------------------------------------------|
|     1.74298x y z  + 1.13265x y z  + 1.31579x y z  - 1.50355x y z  -         |
|             3 1 2           1 3 2           2 1 3           1 2 3           |
|     ------------------------------------------------------------------------|
|     3.35742x y  + 3.07591x y  + 3.83651x y  - 3.71822x y  - 1.99884x y  +   |
|             2 1           3 1           1 2           3 2           1 3     |
|     ------------------------------------------------------------------------|
|     2.11451x y  - .0493297x z  + .0451934x z  - 1.2983y z  + .67642y z  +   |
|             2 3            2 1            3 1          2 1          3 1     |
|     ------------------------------------------------------------------------|
|     .0563687x z  - .0546307x z  + 1.2983y z  - .817669y z  - .0389852x z  + |
|              1 2            3 2          1 2           3 2            1 3   |
|     ------------------------------------------------------------------------|
|     .0412412x z  - .897916y z  + 1.08542y z  - 9.85984x  + 1.09273x  +      |
|              2 3           1 3           2 3           1           2        |
|     ------------------------------------------------------------------------|
|     7.70206x  + .240411y  + .0363213y  - 10.5808y  - 37.2647z  - 5.62995z  +|
|             3           1            2           3           1           2  |
|     ------------------------------------------------------------------------|
|     21.2116z  - .695579, - .981142x y z  + .557965x y z  + .981142x y z  -  |
|             3                      3 2 1           2 3 1           3 1 2    |
|     ------------------------------------------------------------------------|
|     .637583x y z  - .740673x y z  + .846363x y z  - 1.88369x y  +           |
|             1 3 2           2 1 3           1 2 3           2 1             |
|     ------------------------------------------------------------------------|
|     1.72575x y  + 2.15249x y  - 2.08612x y  - 1.12146x y  + 1.18635x y  -   |
|             3 1           1 2           3 2           1 3           2 3     |
|     ------------------------------------------------------------------------|
|     2.14929x z  + 1.96908x z  - 1.39179y z  + .725132y z  + 2.45598x z  -   |
|             2 1           3 1           2 1           3 1           1 2     |
|     ------------------------------------------------------------------------|
|     2.38026x z  + 1.39179y z  - .876553y z  - 1.69859x z  + 1.79688x z  -   |
|             3 2           1 2           3 2           1 3           2 3     |
|     ------------------------------------------------------------------------|
|     .962579y z  + 1.16358y z  + .527434x  + .943204x  + .344042x  -         |
|             1 3           2 3           1           2           3           |
|     ------------------------------------------------------------------------|
|     .522684y  - .543486y  + .0767906y  + 1.04328z  + 1.0848z  + .649757z  + |
|             1           2            3           1          2           3   |
|     ------------------------------------------------------------------------|
|     .838949, 1.4167x  + .705867}                                            |
|                     2                                                       |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use rbpl24s:
====================

  * "rbpl24s(Ring)"

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

The object rbpl24s, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/rbpl24s.m2:46:0.

reimer5 -- 5-dimensional Reimer system
**************************************

  * Usage:
        reimer5(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 146 solutions found in 6.540 seconds (with a Bezout bound of 720).

Reference: PoSSo test suite.

See also: http://homepages.math.uic.edu/~jan/Demo/reimer5.html.

+-----------------------------------------------------------------------------+
|i1 : reimer5(QQ)                                                             |
|                                                                             |
|        2     2     2     2     2        3     3     3     3     3        4  |
|o1 = {2x  - 2y  + 2z  - 2t  + 2u  - 1, 2x  - 2y  + 2z  - 2t  + 2u  - 1, 2x  -|
|     ------------------------------------------------------------------------|
|       4     4     4     4        5     5     5     5     5        6     6   |
|     2y  + 2z  - 2t  + 2u  - 1, 2x  - 2y  + 2z  - 2t  + 2u  - 1, 2x  - 2y  + |
|     ------------------------------------------------------------------------|
|       6     6     6                                                         |
|     2z  - 2t  + 2u  - 1}                                                    |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use reimer5:
====================

  * "reimer5(Ring)"

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

The object reimer5, is a method function (Macaulay2Doc).

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

The source of this document is in ExampleSystems/reimer5.m2:41:0.

rose -- the system Rose, a general economic equilibrium model
*************************************************************

  * Usage:
        rose(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 136 solutions found in 4.954 seconds (with a Bezout bound of 216).

Reference: PoSSo test suite.

See also: http://homepages.math.uic.edu/~jan/Demo/rose.html.

+-----------------------------------------------------------------------------+
|i1 : rose(QQ)                                                                |
|                                                                             |
|       4   20 2   2 4    7   4    7 4   50 2   35     49  3 6 2     5 3      |
|o1 = {y  - --x , x z  + --x*z  + --z  - --x  - --x - ---, -x y z + x y  +    |
|            7           10       48     27     27    216  5                  |
|     ------------------------------------------------------------------------|
|     3 5 2    7 4 3    7 4   2    3 4 3    609 3 3    63 3 2     77 3   2    |
|     -x y z + -x y  - --x y*z  - --x z  + ----x y  + ---x y z - ---x y*z  -  |
|     7        5       20         20       1000       200        125          |
|     ------------------------------------------------------------------------|
|     21 3 3    49  2 3    147 2 2    23863 2   2    91 2 3    27391   3      |
|     --x z  + ----x y  + ----x y z - -----x y*z  - ---x z  - ------x*y  +    |
|     50       1250       2000        60000         400       800000          |
|     ------------------------------------------------------------------------|
|      4137    2    1078     2    5887    3    1029  3    24353    2          |
|     ------x*y z - ----x*y*z  - ------x*z  - ------y  - -------y*z  -        |
|     800000        9375         200000       160000     1920000              |
|     ------------------------------------------------------------------------|
|       343  3                                                                |
|     ------z }                                                               |
|     128000                                                                  |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use rose:
=================

  * "rose(Ring)"

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

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

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

The source of this document is in ExampleSystems/rose.m2:39:0.

sendra -- the system sendra
***************************

  * Usage:
        sendra(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 46 solutions found in 0.111 seconds (with a Bezout bound of 49).

Reference: PoSSo test suite.

See also: http://homepages.math.uic.edu/~jan/Demo/sendra.html.

+-----------------------------------------------------------------------------+
|i1 : sendra(QQ)                                                              |
|                                                                             |
|            4 3         4         3            7       6        5 2       4 3|
|o1 = {- 270x y  - 314x*y  - 689x*y  + 1428, 36x  + 417x y - 422x y  - 270x y |
|     ------------------------------------------------------------------------|
|            3 4        2 5         6       6       5        4 2       3 3    |
|     + 1428x y  - 1475x y  + 510x*y  - 200x  - 174x y - 966x y  + 529x y  +  |
|     ------------------------------------------------------------------------|
|         2 4        5       6       4         2 3         4       5      4   |
|     269x y  + 49x*y  - 267y  + 529x y + 1303x y  - 314x*y  + 262y  + 36x  - |
|     ------------------------------------------------------------------------|
|         2 2         3       4                                               |
|     788x y  - 689x*y  + 177y }                                              |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use sendra:
===================

  * "sendra(Ring)"

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

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

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

The source of this document is in ExampleSystems/sendra.m2:38:0.

trinks -- system of Trinks from the PoSSo test suite
****************************************************

  * Usage:
        trinks(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 10 solutions found in 0.203 seconds (with a Bezout bound of 24).

Reference: See the PoSSo test suite.

See also: http://homepages.math.uic.edu/~jan/Demo/trinks.html.

+-----------------------------------------------------------------------------+
|i1 : trinks(QQ)                                                              |
|                                                                             |
|                                                                2            |
|o1 = {45y + 35u - 165v - 36, 35y + 25z + 40t - 27u, 25y*u - 165v  + 15x - 18z|
|     ------------------------------------------------------------------------|
|                                     3                           2           |
|     + 30t, 15y*z + 20t*u - 9x, - 11v  + x*y + 2z*t, - 11u*v + 3v  + 99x}    |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use trinks:
===================

  * "trinks(Ring)"

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

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

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

The source of this document is in ExampleSystems/trinks.m2:44:0.

virasoro -- the construction of Virasoro algebras
*************************************************

  * Usage:
        virasoro(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 256 solutions found in 0.356 seconds (with a Bezout bound of 256).

Reference: "Generalized Virasoro Constructions for SU(3)" by S. Schrans and W.
Troost (pages 584-606).

See also: http://homepages.math.uic.edu/~jan/Demo/virasoro.html.

+-----------------------------------------------------------------------------+
|i1 : virasoro(QQ)                                                            |
|                                                                             |
|        2                                                                    |
|o1 = {8x  + 8x x  + 8x x  - 8x x  + 2x x  + 2x x  + 2x x  - 2x x  + 2x x  -  |
|        1     1 2     1 3     2 3     1 4     1 5     1 6     5 6     1 7    |
|     ------------------------------------------------------------------------|
|                           2                                                 |
|     2x x  - x , 8x x  + 8x  - 8x x  + 8x x  + 2x x  + 2x x  + 2x x  - 2x x  |
|       4 7    1    1 2     2     1 3     2 3     2 4     2 5     2 6     4 6 |
|     ------------------------------------------------------------------------|
|                                                       2                     |
|     + 2x x  - 2x x  - x , - 8x x  + 8x x  + 8x x  + 8x  + 2x x  + 2x x  -   |
|         2 7     5 7    2      1 2     1 3     2 3     3     3 4     3 5     |
|     ------------------------------------------------------------------------|
|                                                                   2         |
|     2x x  + 2x x  + 2x x  - 2x x  - x , 2x x  + 2x x  + 2x x  + 8x  - 2x x  |
|       4 5     3 6     3 7     6 7    3    1 4     2 4     3 4     4     3 5 |
|     ------------------------------------------------------------------------|
|                                                                             |
|     + 8x x  - 2x x  + 2x x  - 2x x  + 2x x  + 6x x  - 6x x  - x , - 2x x  + |
|         4 5     2 6     4 6     1 7     4 7     4 8     5 8    4      3 4   |
|     ------------------------------------------------------------------------|
|                                       2                                     |
|     2x x  + 2x x  + 2x x  + 8x x  + 8x  - 2x x  + 2x x  - 2x x  + 2x x  -   |
|       1 5     2 5     3 5     4 5     5     1 6     5 6     2 7     5 7     |
|     ------------------------------------------------------------------------|
|                                                                             |
|     6x x  + 6x x  - x , - 2x x  - 2x x  + 2x x  + 2x x  + 2x x  + 2x x  +   |
|       4 8     5 8    5      2 4     1 5     1 6     2 6     3 6     4 6     |
|     ------------------------------------------------------------------------|
|               2                                                             |
|     2x x  + 8x  - 2x x  + 8x x  + 6x x  - 6x x  - x , - 2x x  - 2x x  -     |
|       5 6     6     3 7     6 7     6 8     7 8    6      1 4     2 5       |
|     ------------------------------------------------------------------------|
|                                                               2             |
|     2x x  + 2x x  + 2x x  + 2x x  + 2x x  + 2x x  + 8x x  + 8x  - 6x x  +   |
|       3 6     1 7     2 7     3 7     4 7     5 7     6 7     7     6 8     |
|     ------------------------------------------------------------------------|
|                                                                     2       |
|     6x x  - x , - 6x x  - 6x x  + 6x x  + 6x x  + 6x x  + 6x x  + 8x  - x } |
|       7 8    7      4 5     6 7     4 8     5 8     6 8     7 8     8    8  |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use virasoro:
=====================

  * "virasoro(Ring)"

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

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

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

The source of this document is in ExampleSystems/virasoro.m2:46:0.

wood -- system derived from optimizing the Wood function
********************************************************

  * Usage:
        wood(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 9 solutions found in 0.141 seconds (with a Bezout bound of 36).

Reference: "Testing unconstrained optimization software" by J.J. More, B.S.
Garbow and K.E. Hilstrom (pages 17-41).

See also: http://homepages.math.uic.edu/~jan/Demo/wood.html.

+-----------------------------------------------------------------------------+
|i1 : wood(RR_53)                                                             |
|                                                                             |
|          3                           2                             3        |
|o1 = {200x  - 200x x  + x  - 1, - 100x  + 110.1x  + 9.9x  - 20, 180x  -      |
|          1       1 2    1            1         2       4           3        |
|     ------------------------------------------------------------------------|
|                            2                                                |
|     180x x  + x  - 1, - 90x  + 9.9x  + 100.1x  - 20}                        |
|         3 4    3           3       2         4                              |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use wood:
=================

  * "wood(Ring)"

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

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

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

The source of this document is in ExampleSystems/wood.m2:42:0.

wright -- system of A.H. Wright
*******************************

  * Usage:
        wright(kk)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the coefficient ring
  * Outputs:
      * a list (Macaulay2Doc), of the polynomials in the system

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

This system was solved in May 2020, using solveSystem (NumericalAlgebraicGeometry),
in Macaulay2 v1.15 with an
Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 32 solutions found in 0.0944 seconds (with a Bezout bound of 32).

Reference: "Computation of all solutions to a system of polynomial equations"
by M. Kojima and S. Mizuno (pages 131-157), "Finding all solutions to a system
of polynomial equations" by A.H. Wright (pages 125-133), and "A simple homotopy
method for determining all isolated solutions to polynomial systems" by W.
Zulehner (pages 167-177).

See also: http://homepages.math.uic.edu/~jan/Demo/wright.html.

+-----------------------------------------------------------------------------+
|i1 : wright(QQ)                                                              |
|                                                                             |
|       2                                 2                                 2 |
|o1 = {x  - x  + x  + x  + x  + x  - 10, x  + x  - x  + x  + x  + x  - 10, x  |
|       1    1    2    3    4    5        2    1    2    3    4    5        3 |
|     ------------------------------------------------------------------------|
|                                     2                                 2     |
|     + x  + x  - x  + x  + x  - 10, x  + x  + x  + x  - x  + x  - 10, x  + x |
|        1    2    3    4    5        4    1    2    3    4    5        5    1|
|     ------------------------------------------------------------------------|
|     + x  + x  + x  - x  - 10}                                               |
|        2    3    4    5                                                     |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use wright:
===================

  * "wright(Ring)"

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

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

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

The source of this document is in ExampleSystems/wright.m2:45:0.