swMATH ID: 33429
Software Authors: Antonio Macchia, Amy Wiebe
Description: Slack Ideals in Macaulay2. Recently Gouveia, Thomas and the authors introduced the slack realization space, a new model for the realization space of a polytope. It represents each polytope by its slack matrix, the matrix obtained by evaluating each facet inequality at each vertex. Unlike the classical model, the slack model naturally mods out projective transformations. It is inherently algebraic, arising as the positive part of a variety of a saturated determinantal ideal, and provides a new computational tool to study classical realizability problems for polytopes. We introduce the package SlackIdeals for Macaulay2, that provides methods for creating and manipulating slack matrices and slack ideals of convex polytopes and matroids. Slack ideals are often difficult to compute. To improve the power of the slack model, we develop two strategies to simplify computations: we scale as many entries of the slack matrix as possible to one; we then obtain a reduced slack model combining the slack variety with the more compact Grassmannian realization space model. This allows us to study slack ideals that were previously out of computational reach. As applications, we show that the well-known Perles polytope does not admit rational realizations and prove the non-realizability of a large simplicial sphere.
Homepage: https://arxiv.org/abs/2003.07382
Source Code:  https://bitbucket.org/macchia/slackideals/src/master/SlackIdeals.m2
Dependencies: Macaulay2
Keywords: Combinatorics; arXiv_math.CO; arXiv_Commutative Algebra; math.AC; Algebraic Geometry; arXiv_math.AG; Macaulay2; Polytopes; Slack matrices; Slack ideals; Matroids
Related Software: Macaulay2; 2L_enum
Cited in: 2 Documents

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Slack Ideals in Macaulay2
Antonio Macchia, Amy Wiebe

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