Four Dimensional Polytopes of Minimum Positive Semidefinite Rank

TL;DR

This paper classifies all psd-minimal 4-polytopes using new algebraic tools, revealing 31 classes and providing explicit realizations, thereby advancing understanding of the geometric and algebraic structure of these polytopes.

Contribution

It introduces trinomial obstructions and the slack ideal to classify psd-minimal 4-polytopes, completing the classification and identifying counterexamples to previous conjectures.

Findings

31 combinatorial classes of psd-minimal 4-polytopes identified

Explicit psd-minimal realizations provided for each class

Counterexamples to some open conjectures discovered

Abstract

The positive semidefinite (psd) rank of a polytope is the size of the smallest psd cone that admits an affine slice that projects linearly onto the polytope. The psd rank of a d-polytope is at least d+1, and when equality holds we say that the polytope is psd-minimal. In this paper we develop new tools for the study of psd-minimality and use them to give a complete classification of psd-minimal 4-polytopes. The main tools introduced are trinomial obstructions, a new algebraic obstruction for psd-minimality, and the slack ideal of a polytope, which encodes the space of realizations of a polytope up to projective equivalence. Our central result is that there are 31 combinatorial classes of psd-minimal 4-polytopes. We provide combinatorial information and an explicit psd-minimal realization in each class. For 11 of these classes, every polytope in them is psd-minimal, and these are precisely the combinatorial classes of the known projectively unique 4-polytopes. We give a complete characterization of psd-minimality in the remaining classes, encountering in the process counterexamples to some open conjectures.