Four Dimensional Polytopes of Minimum Positive Semidefinite Rank
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 dpolytope is at least d+1, and when equality holds we say that the polytope is psdminimal. In this paper we develop new tools for the study of psdminimality and use them to give a complete classification of psdminimal 4polytopes. The main tools introduced are trinomial obstructions, a new algebraic obstruction for psdminimality, 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 psdminimal 4polytopes. We provide combinatorial information and an explicit psdminimal realization in each class. For 11 of these classes, every polytope in them is psdminimal, and these are precisely the combinatorial classes of the known projectively unique 4polytopes. We give a complete characterization of psdminimality in the remaining classes, encountering in the process counterexamples to some open conjectures.
 Publication:

arXiv eprints
 Pub Date:
 May 2015
 DOI:
 10.48550/arXiv.1506.00187
 arXiv:
 arXiv:1506.00187
 Bibcode:
 2015arXiv150600187G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry;
 Mathematics  Optimization and Control