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SlackIdeals -- a package for slack ideals of polytopes and matroids

Description

SlackIdeals is a package which allows the user to create slack realizations and the slack ideal of a polytope or a matroid. Polytopes and matroids may be entered as a list of vertices of a specific realization or as a pre-created Polyhedron or Matroid object (using the packages Polyhedra and Matroids).

References.

  • [GMTW19] The slack realization space of a polytope, (J. Gouveia, A. Macchia, R.R. Thomas, A. Wiebe, SIAM J. Discrete Math. 33 (2019), 3, 1637–1653.)
  • [BW19] The slack realization space of a matroid, (M. Brandt, A. Wiebe, Algebraic Combinatorics, 2 (2019), 4, 663–681, 2019.)
  • [BMTW20] Projectively unique polytopes and toric slack ideals, (J. Gouveia, A. Macchia, R.R. Thomas, A. Wiebe, J. Pure Appl. Algebra 224 (2020), 5, paper 106229.)
  • [BMW20] Combining realization space models of polytopes, (J. Gouveia, A. Macchia, A. Wiebe, preprint (2020), arXiv:2001.11999v1.)

Authors

Version

This documentation describes version 1.0 of SlackIdeals.

Citation

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

@misc{SlackIdealsSource,
  title = {{SlackIdeals: Slack ideals of polytopes and matroids. Version~1.0}},
  author = {Amy Wiebe and Antonio Macchia},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • containsFlag -- establishes whether or not a list of facet labels contains a flag in a polytope or matroid
    • cycleIdeal -- constructs the cycle ideal of a realization
    • findFlag -- computes a list of facet labels that make up a flag in a polytope
    • getFacetBases -- get a list of d-spanning elements for each facet
    • graphFromSlackMatrix -- creates the vertex-edge incidence matrix for the bipartite non-incidence graph with adjacency matrix the given slack matrix
    • graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope
    • grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • reconstructSlackMatrix -- a list of facet labels that make up a flag in a polytope
    • reducedSlackMatrix -- a reduced slack matrix of a polytope
    • rehomogenizeIdeal -- rehomogenization of a the dehomogenized slack ideal
    • rehomogenizePolynomial -- rehomogenization of a polynomial reversing the dehomogenization of the slack matrix
    • setOnesForest -- sets to 1 variables in a symbolic slack matrix which corresponding to edges of a spanning forest
    • slackFromGaleCircuits -- computes the slack matrix of a polytope from a Gale transform of the polytope
    • slackFromGalePlucker -- fill the slack matrix with Plücker coordinates of the Gale transform
    • slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
    • slackIdeal -- computes the slack ideal
    • slackMatrix -- computes the slack matrix of a given realization
    • specificSlackMatrix -- creates built-in slack matrices of some polytopes and matroids
    • symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • toricPolytope -- computes the polytope whose toric ideal is the given ideal
    • universalIdeal -- computes the universal realization ideal of a matroid
  • Methods
    • containsFlag(List,List) -- see containsFlag -- establishes whether or not a list of facet labels contains a flag in a polytope or matroid
    • containsFlag(List,Matrix) -- see containsFlag -- establishes whether or not a list of facet labels contains a flag in a polytope or matroid
    • containsFlag(List,Matroid) -- see containsFlag -- establishes whether or not a list of facet labels contains a flag in a polytope or matroid
    • containsFlag(List,Polyhedron) -- see containsFlag -- establishes whether or not a list of facet labels contains a flag in a polytope or matroid
    • cycleIdeal(List) -- see cycleIdeal -- constructs the cycle ideal of a realization
    • cycleIdeal(Matrix) -- see cycleIdeal -- constructs the cycle ideal of a realization
    • cycleIdeal(Matroid) -- see cycleIdeal -- constructs the cycle ideal of a realization
    • cycleIdeal(Polyhedron) -- see cycleIdeal -- constructs the cycle ideal of a realization
    • findFlag(List) -- see findFlag -- computes a list of facet labels that make up a flag in a polytope
    • findFlag(Matrix) -- see findFlag -- computes a list of facet labels that make up a flag in a polytope
    • findFlag(Matroid) -- see findFlag -- computes a list of facet labels that make up a flag in a polytope
    • findFlag(Polyhedron) -- see findFlag -- computes a list of facet labels that make up a flag in a polytope
    • getFacetBases(List) -- see getFacetBases -- get a list of d-spanning elements for each facet
    • getFacetBases(Matrix) -- see getFacetBases -- get a list of d-spanning elements for each facet
    • getFacetBases(Matroid) -- see getFacetBases -- get a list of d-spanning elements for each facet
    • getFacetBases(Polyhedron) -- see getFacetBases -- get a list of d-spanning elements for each facet
    • graphFromSlackMatrix(Matrix) -- see graphFromSlackMatrix -- creates the vertex-edge incidence matrix for the bipartite non-incidence graph with adjacency matrix the given slack matrix
    • graphicIdeal(List) -- see graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope
    • graphicIdeal(Matrix) -- see graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope
    • graphicIdeal(Matroid) -- see graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope
    • graphicIdeal(Polyhedron) -- see graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope
    • grassmannSectionIdeal(Cone) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(List) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(List,List) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(Matrix) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(Matrix,List) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(Matroid) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • grassmannSectionIdeal(Polyhedron) -- see grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix
    • reconstructSlackMatrix(Matrix,List) -- see reconstructSlackMatrix -- a list of facet labels that make up a flag in a polytope
    • reconstructSlackMatrix(Matrix,List,List) -- see reconstructSlackMatrix -- a list of facet labels that make up a flag in a polytope
    • reducedSlackMatrix(List) -- see reducedSlackMatrix -- a reduced slack matrix of a polytope
    • reducedSlackMatrix(Matrix) -- see reducedSlackMatrix -- a reduced slack matrix of a polytope
    • reducedSlackMatrix(ZZ,Matrix) -- see reducedSlackMatrix -- a reduced slack matrix of a polytope
    • rehomogenizeIdeal(ZZ,Matrix) -- see rehomogenizeIdeal -- rehomogenization of a the dehomogenized slack ideal
    • rehomogenizeIdeal(ZZ,Matrix,Graph) -- see rehomogenizeIdeal -- rehomogenization of a the dehomogenized slack ideal
    • rehomogenizePolynomial(Matrix) -- see rehomogenizePolynomial -- rehomogenization of a polynomial reversing the dehomogenization of the slack matrix
    • rehomogenizePolynomial(Matrix,Matrix,Graph,RingElement) -- see rehomogenizePolynomial -- rehomogenization of a polynomial reversing the dehomogenization of the slack matrix
    • setOnesForest(Matrix) -- see setOnesForest -- sets to 1 variables in a symbolic slack matrix which corresponding to edges of a spanning forest
    • slackFromGaleCircuits(Matrix) -- see slackFromGaleCircuits -- computes the slack matrix of a polytope from a Gale transform of the polytope
    • slackFromGalePlucker(List,List) -- see slackFromGalePlucker -- fill the slack matrix with Plücker coordinates of the Gale transform
    • slackFromGalePlucker(List,Matrix) -- see slackFromGalePlucker -- fill the slack matrix with Plücker coordinates of the Gale transform
    • slackFromPlucker(List) -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
    • slackFromPlucker(List,List) -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
    • slackFromPlucker(Matroid) -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
    • slackFromPlucker(Polyhedron) -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
    • slackIdeal(Cone) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(List) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(Matrix) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(Matroid) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(Polyhedron) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(ZZ,List) -- see slackIdeal -- computes the slack ideal
    • slackIdeal(ZZ,Matrix) -- see slackIdeal -- computes the slack ideal
    • slackMatrix(Cone) -- see slackMatrix -- computes the slack matrix of a given realization
    • slackMatrix(List) -- see slackMatrix -- computes the slack matrix of a given realization
    • slackMatrix(Matroid) -- see slackMatrix -- computes the slack matrix of a given realization
    • slackMatrix(Polyhedron) -- see slackMatrix -- computes the slack matrix of a given realization
    • specificSlackMatrix(String) -- see specificSlackMatrix -- creates built-in slack matrices of some polytopes and matroids
    • symbolicSlackMatrix(Cone) -- see symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackMatrix(List) -- see symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackMatrix(Matrix) -- see symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackMatrix(Matroid) -- see symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackMatrix(Polyhedron) -- see symbolicSlackMatrix -- computes the symbolic slack matrix
    • symbolicSlackOfPlucker(List) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(List,List) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(Matrix) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(Matrix,List) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(Matroid) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(Polyhedron) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • symbolicSlackOfPlucker(ZZ,List) -- see symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
    • toricPolytope(Ideal) -- see toricPolytope -- computes the polytope whose toric ideal is the given ideal
    • universalIdeal(List) -- see universalIdeal -- computes the universal realization ideal of a matroid
    • universalIdeal(Matroid) -- see universalIdeal -- computes the universal realization ideal of a matroid
  • Symbols
    • FlagElement -- a facet label that will be contained in a flag of facets of given polytope or matroid
    • FlagIndices -- a list of facet labels that form a flag of facets of given polytope or matroid
    • Object -- select the combinatorial object which the input should be interpreted as
    • Saturate -- choose whether to saturate with respect to the product of all variables at the same time or variable by variable.
    • Tolerance -- choose the tolerance to approximate computations over the field RR
    • Vars -- give a set of variables for the polynomial ring where the object created will live

For the programmer

The object SlackIdeals is a package, defined in SlackIdeals.m2.

The source of this document is in SlackIdeals.m2:1505:0.