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arXiv · 1506.00187

Four Dimensional Polytopes of Minimum Positive Semidefinite Rank

João Gouveia·Kanstanstin Pashkovich·Richard Z. Robinson·Rekha R. Thomas
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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.

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João Gouveia, Kanstanstin Pashkovich, Richard Z. Robinson, Rekha R. Thomas. 2015-05-31. Four Dimensional Polytopes of Minimum Positive Semidefinite Rank. https://arxiv.org/abs/1506.00187

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