Four-dimensional polytopes of minimum positive semidefinite rank
Four-dimensional polytopes of minimum positive semidefinite rank
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DOI：
10.1016/j.jcta.2016.08.002
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发表时间：
2017-01-01
影响因子：
1.1
通讯作者：
Thomas, Rekha R.
中科院分区：
文献类型：
--
作者：
Gouveia, Joao;Pashkovich, Kanstanstin;Thomas, Rekha R.
一个多面体的半正定（psd）秩是最小psd锥的大小，该锥允许线性投影到该多面体上的一个线性切片。d-多面体的psd秩至少是d + 1，当等式成立时，我们说这个多面体是psd-极小的。在本文中，我们开发了新的工具来研究psd-极小性，并利用它们给出了psd-极小4-多面体的一个完整分类。主要的工具是三项障碍，一个新的代数障碍psd-极小，和松弛理想的多面体，它编码的实现空间的多面体到射影等价。我们的中心结果是，有31个组合类的psd-极小4-多面体。我们提供组合信息和明确的PSD最小实现在每个类。对于其中的11个类，它们中的每个多面体都是psd-极小的，并且这些正是已知的射影唯一4-多面体的组合类。我们给出了一个完整的表征PSD-极小在剩余的类，遇到的过程中的反例，一些开放的approximatures。(C)2016 Elsevier Inc. All rights reserved.
The positive semidefinite (psd) rank of a polytope is the size of the smallest psd cone that admits an aifine 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-rninimal 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. (C) 2016 Elsevier Inc. All rights reserved.