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SlackIdeals :: SlackIdeals

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.

Source code

The source code from which this documentation is derived is in the file SlackIdeals.m2.

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 Plucker coordinates of the Gale transform
    • slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker 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 Plucker 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 Plucker coordinates of the Gale transform
    • "slackFromGalePlucker(List,Matrix)" -- see slackFromGalePlucker -- fill the slack matrix with Plucker coordinates of the Gale transform
    • "slackFromPlucker(List)" -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker coordinates
    • "slackFromPlucker(List,List)" -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker coordinates
    • "slackFromPlucker(Matroid)" -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker coordinates
    • "slackFromPlucker(Polyhedron)" -- see slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker 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 Plucker variables
    • "symbolicSlackOfPlucker(List,List)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker variables
    • "symbolicSlackOfPlucker(Matrix)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker variables
    • "symbolicSlackOfPlucker(Matrix,List)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker variables
    • "symbolicSlackOfPlucker(Matroid)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker variables
    • "symbolicSlackOfPlucker(Polyhedron)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker variables
    • "symbolicSlackOfPlucker(ZZ,List)" -- see symbolicSlackOfPlucker -- fill the slack matrix with Plucker 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.