期刊
JOURNAL OF PURE AND APPLIED ALGEBRA
卷 224, 期 5, 页码 -出版社
ELSEVIER
DOI: 10.1016/j.jpaa.2019.106229
关键词
Polytopes; Slack matrix; Slack ideal; Realization spaces; Toric ideal; Projectively unique polytopes
资金
- FCT through CMUC [UID/MAT/00324/2013]
- FCT through P2020 [UID/MAT/00324/2013, SAICTPAC/0011/2015]
- INdAM
- U.S. National Science Foundation [DMS-1418728, DMS-1719538]
- NSERC
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.
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