The Slack Realization Space of a Polytope

TL;DR

This paper introduces the slack realization space of a polytope, a new algebraic model that encodes polytope realizability and combinatorics, providing tools for classical geometric questions.

Contribution

It presents a novel algebraic framework for the realization space of polytopes using slack ideals, connecting combinatorics and algebraic geometry.

Findings

Defines the slack realization space via positive parts of algebraic varieties.

Provides an effective computational approach for polytope realizability questions.

Establishes connections between slack ideals and classical polytope problems.

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.