Four-dimensional polytopes of minimum positive semidefinite rank

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• 作者：Joã ; o Gouveia^a ; ^{jgouveia@mat.uc.pt" class="auth_mail" title="E-mail the corresponding author} ; Kanstanstin Pashkovich^b ; ^{kanstantsin.pashkovich@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Richard Z. Robinson^c ; ^{rzr@uw.edu" class="auth_mail" title="E-mail the corresponding author} ; Rekha R. Thomas^c ; ^{rrthomas@uw.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polytopes ; Positive semidefinite rank ; psd-minimal ; Slack matrix ; Slack ideal
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：184-226
• 全文大小：940 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300747&_mathId=si1.gif&_user=111111111&_pii=S0097316516300747&_rdoc=1&_issn=00973165&md5=d9eb65dbff49e37ede1f981861e424c6" title="Click to view the MathML source">d+1class="mathContainer hidden">class="mathCode">$d+1$, 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.