next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 -- given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, creates a sorted list of vertices (empty if a matrix is given as input) in the order corresponding to B, and B the 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 -- given a slack matrix of a polytope, a cone or a matroid, or a set of polytope vertices, cone generators, or matroid vectors, and a set of set of hyperplane spanning set indices, it computes the Grassmannian section ideal corresponding to choice B of the object with slack matrix S
    • 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 -- given a set of vectors of a Gale transform or a matrix whose columns form a Gale transform of a polytope, it fills the slack matrix of the polytope with Plucker coordinates of the Gale transform
    • slackFromPlucker -- given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, it fills the corresponding slack matrix 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 -- given the number of polytope vertices, cone generators, or matroid vectors, or a set of polytope vertices, cone generators, or matroid vectors, or a slack matrix and a set of set of hyperplane spanning set indices, it fills 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
  • 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