Interlacing Ehrhart polynomials of reflexive polytopes

Higashitani, Akihiro; Kummer, Mario; Michałek, Mateusz

doi:10.1007/s00029-017-0350-6

Cited by 25 publications

(39 citation statements)

References 29 publications

(38 reference statements)

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“…Our result generalizes [17,Proposition 4.4], where the case min(a, b) ≤ 3 was studied. From the formula it is apparent that the h * -polynomial is palindromic, that is, t a+b+1 h * a,b ( 1 t ) = h * a,b (t) which reflects that P K a+1,b+1 is reflexive by a famous theorem of Hibi [14].…”

Section: Introductionsupporting

confidence: 77%

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Arithmetic Aspects of Symmetric Edge Polytopes

2019

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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).

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Section: Introductionsupporting

confidence: 77%

“…A recursive formula. In [17] recursive formulas for h * a,b were given for any fixed a ≤ 2. These formulas played a fundamental role in the study of the roots of the Ehrhart polynomial of P K a+1,b+1 .…”

Section: 7mentioning

confidence: 99%

Arithmetic Aspects of Symmetric Edge Polytopes

2019

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“…16,19 Similar constructions have also appeared in other contexts. [20][21][22] The adjacency polytope bound is a simplification and relaxation of the Bernshtein-Kushnirenko-Khovanskii bound. 23 Recently, the author proved that the this bound remains a sharp upper bound for the total number of complex synchronization configurations for certain graphs.…”

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

Directed acyclic decomposition of Kuramoto equations

Chen

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divideand-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.The spontaneous synchronization of a network of oscillators is an emergent phenomenon that naturally appears in many seemingly independent complex systems including mechanical, chemical, biological, and even social systems. The Kuramoto model is one of the most widely studied and successful mathematical models for analyzing synchronization behaviors. While much is known about the macro-scale question of whether or not a Kuramoto network can be synchronized, detailed analysis of the possible configurations of the oscillator once it has reached synchronization remains difficult for large networks partly due to the nonlinear interactions involved. In this work, we demonstrate that by dividing the link between two oscillators into two one-way interactions, complex networks can indeed be decomposed into much simpler subnetwork. This is a crucial step toward fully understanding synchronization configurations in large networks. arXiv:1903.04492v2 [math.CO]

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“…In fact, the Ehrhart polynomials of special symmetric edge polytopes have properties similar to Riemann's ζ function [5,29]. Moreover, many results about zero loci of the Ehrhart polynomials of symmetric edge polytopes have been found from a viewpoint of algebraic combinatorics [19,22,25].…”

Section: Introductionmentioning

confidence: 99%

Symmetric edge polytopes and matching generating polynomials

Ohsugi¹,

Tsuchiya²

2021

Combinatorial Theory

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Symmetric edge polytopes A G of type A are lattice polytopes arising from the root system A n and finite simple graphs G. There is a connection between A G and the Kuramoto synchronization model in physics. In particular, the normalized volume of A G plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph G, we give a formula for the h * -polynomial of A G by using matching generating polynomials, where G is the suspension of G. This gives also a formula for the normalized volume of A G . Moreover, via methods from chemical graph theory, we show that for any cactus graph G, the h * -polynomial of A G is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type B, which are lattice polytopes arising from the root system B n and finite simple graphs.

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Interlacing Ehrhart polynomials of reflexive polytopes

Cited by 25 publications

References 29 publications

Arithmetic Aspects of Symmetric Edge Polytopes

Arithmetic Aspects of Symmetric Edge Polytopes

Directed acyclic decomposition of Kuramoto equations

Symmetric edge polytopes and matching generating polynomials

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