Projectively unique polytopes and toric slack ideals
Projectively unique polytopes and toric slack ideals
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DOI：
10.1016/j.jpaa.2019.106229
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发表时间：
2020-05-01
影响因子：
0.8
通讯作者：
Wiebe, Amy
中科院分区：
文献类型：
--
作者：
Gouveia, Joao;Macchia, Antonio;Wiebe, Amy
多胞形的松弛理想是一个饱和的行列式理想，它给出了多胞形实现空间的一个新模型。最简单的松弛理想是环面的，并且与射影唯一多面体有联系。证明了如果一个射影唯一的多面体有一个环面松弛理想，则它是该多面体的顶点-刻面不关联二部图的环面理想。一个多面体的松弛理想包含在这个环面理想中当且仅当这个多面体是道德上的2-层次，这是多面体中2-层次性质的推广。我们表明，不承认理性实现的多面体不能有环面松弛理想。一个经典的例子，一个射影唯一的多面体没有合理的实现是由于珀尔。我们证明了松弛理想的Perles多面体是可约的，提供了第一个例子的松弛理想，是不是素数。(C)2019 Elsevier B. V.版权所有。
The slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. The simplest slack ideals are toric and have connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non-incidences of the polytope. The slack ideal of a polytope is contained in this toric ideal if and only if the polytope is morally 2-level, a generalization of the 2-level property in polytopes. We show that polytopes that do not admit rational realizations cannot have toric slack ideals. A classical example of a projectively unique polytope with no rational realizations is due to Perles. We prove that the slack ideal of the Perles polytope is reducible, providing the first example of a slack ideal that is not prime. (C) 2019 Elsevier B.V. All rights reserved.