We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacings polynomials we provide an explicit formula for the h * -polynomial in case of complete bipartite graphs. In particular, we show that the h * -polynomial is γ-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).2010 Mathematics Subject Classification. 05A15, 52B12 (primary); 13P10, 26C10, 52B15, 52B20 (secondary).
We analyze tensors in C m ⊗C m ⊗C m satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of End(C m ) and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions.Sommaire. Nousétudions des tenseurs dans C m ⊗C m ⊗C m satisfaisant leséquations de Strassen lorsque le rang du bord vaut m. Les résultats obtenus comprennent : deux caractérisations purement géométriques du tenseur de Coppersmith-Winograd, une réductionà l'étude des tenseurs symétriques sous une hypothèse raisonnable de généricité, et beaucoup de nouveaux exemples etéquations. Cetteétude est liée de prèsà l'étude de la variété des sous-espaces abéliens de dimension m de End(C m ) et la sous-variété obtenue comme l'adhérence de Zariski de la variété des tores maximaux, appelée variété des réductions. T , denoted R(T ) is the smallest r such that T may be expressed as the sum of r rank one tensors. Rank is not semi-continuous, so one defines the border rank of T , denoted R(T ), to be the smallest r such that T is a limit of tensors of rank r, or equivalently (see e.g. [27, Cor. 5.1.1.5]) the smallest r such that T lies in the Zariski closure of the set of tensors of rank r.• A class of tensors for which Strassen's additivity conjecture holds (Theorem 4.1).1.1. Background and previous work. The maximum rank of T ∈ C m ⊗C m ⊗C m is not known, it is easily seen to be at most m 2 (and known to be at most 2 3 m 2 [7]), and of course is at least the maximum border rank. The maximum border rank is ⌈ m 3 −1 3m−2 ⌉ except when m = 3 when it is five [34,40]. In computer science, there is interest in producing explicit tensors of high rank and border rank. The maximal rank of a known explicit tensor is 3m − log 2 (m) − 3 when m is a power of two [1], see Example 3.3.Tensors in A⊗B⊗C are completely understood when all vector spaces have dimension at most three [11]. In particular, for tensors of border rank three, the maximum rank is five. The case of C 2 ⊗C m ⊗C m is also completely understood, see, e.g. [27, §10.3]. While tensors of border rank 4 in C 4 ⊗C 4 ⊗C 4 are essentially understood [19,20,5], the ranks of such tensors are not known. We determine their ranks under our two genericity hypotheses. The difficulty of understanding border rank four tensors in C 4 ⊗C 4 ⊗C 4 (which was first overcome in [19]) was non-concision, which we avoid in this paper.Let Red 0,SL (m) denote the set of all maximal tori in SL(m), i.e., the set of all (m − 1)dimensional abelian subgroups that are diagonalizable. It can be given a topology (called the Chabauty topology, see [33]) and its closure Red SL (m) is studied in [33]. (A. Leitner works over R, but this changes little.) If one considers the corresponding L...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.