1. The Slack Realization Space of a Polytope
- Author
-
Amy Wiebe, Rekha R. Thomas, João Gouveia, and Antonio Macchia
- Subjects
General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Polytope ,0102 computer and information sciences ,Commutative Algebra (math.AC) ,Space (mathematics) ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Realizability ,FOS: Mathematics ,Mathematics - Combinatorics ,Mathematics::Metric Geometry ,Algebraic Geometry (math.AG) ,Computer Science::Operating Systems ,Equivalence (measure theory) ,Mathematics ,Discrete mathematics ,Mathematics::Combinatorics ,Mathematics::Commutative Algebra ,Mathematics - Commutative Algebra ,010201 computation theory & mathematics ,Combinatorics (math.CO) ,Realization (systems) ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes., Comment: The original arXiv submission of this paper has now been split into two parts; this paper describing the slack realization space of a polytope and applications of the slack ideal, and a second paper focussing on toric slack ideals and projective uniqueness
- Published
- 2019