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https://hdl.handle.net/10316/44073
Title: | Four-dimensional polytopes of minimum positive semidefinite rank | Authors: | Gouveia, João Pashkovich, Kanstanstin Robinson, Richard Z. Thomas, Rekha R. |
Issue Date: | 2017 | Publisher: | Elsevier | Project: | info:eu-repo/grantAgreement/FCT/5876/147205/PT | Serial title, monograph or event: | Journal of Combinatorial Theory, Series A | Volume: | 145 | 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. | URI: | https://hdl.handle.net/10316/44073 | DOI: | 10.1016/j.jcta.2016.08.002 10.1016/j.jcta.2016.08.002 |
Rights: | embargoedAccess |
Appears in Collections: | I&D CMUC - Artigos em Revistas Internacionais |
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