Roots of Ehrhart polynomials arising from graphs

Matsui, Tetsushi; Higashitani, Akihiro; Nagazawa, Yuuki; Ohsugi, Hidefumi; Hibi, Takayuki

doi:10.1007/s10801-011-0290-8

Cited by 48 publications

(63 citation statements)

References 20 publications

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“…As we have seen in the previous section, OPS provide strong methods to prove that zeros of polynomials lie on R. However, as conjectured, e.g., in [24,Conj. 4.7], there exist families with roots on R, but not giving rise to OPS.…”

Section: Ehrhart Polynomials and Bipartite Graphsmentioning

confidence: 85%

“…Furthermore, it is terminal if every lattice point on the boundary is a vertex. For (nonoriented) graphs G the polytope P G is reflexive and terminal [24,Prop. 4.2].…”

Section: Graphs Polytopes and Polynomialsmentioning

confidence: 99%

“…There are known constructions that associate to a graph a lattice polytope-the symmetric edge polytope [24,27,28]. Furthermore, to each lattice polytope P one associates the Ehrhart polynomial H P [8,11,15,31] that computes the number of lattice points in dilations of P. This is also the Hilbert polynomial of the normal toric variety Spec C [C], where C is the cone over P [14,32].…”

Section: Introductionmentioning

confidence: 99%

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Interlacing Ehrhart polynomials of reflexive polytopes

Higashitani

Kummer

Michałek

2017

Sel. Math. New Ser.

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It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties shared by the Riemann ζ function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.Mathematics Subject Classification Primary 26C10; Secondary 52B20 · 12D10 · 30C15 · 05C31 · 33C45 · 65H04

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Section: Ehrhart Polynomials and Bipartite Graphsmentioning

confidence: 85%

“…Furthermore, it is terminal if every lattice point on the boundary is a vertex. For (nonoriented) graphs G the polytope P G is reflexive and terminal [24,Prop. 4.2].…”

Section: Graphs Polytopes and Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Interlacing Ehrhart polynomials of reflexive polytopes

Higashitani

Kummer

Michałek

2017

Sel. Math. New Ser.

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“…Now, we study the function h(y) = It has been conjectured that all roots a ∈ C of i(P ± G , n) satisfy (a) = −1/2 in [16].…”

Section: Is a Root Of G(y) If And Only Ifmentioning

confidence: 99%

“…Write P ± G ⊂ R d for the convex polytope which is the convex hull of {ρ(e) : e ∈ E(G)} ∪ {μ(e) : e ∈ E(G)}. Let H ⊂ R d denote the hyperplane defined by the equation [16,Proposition 3.2].) One of the research problems is to find a combinatorial characterization of the finite graphs G for which all roots a ∈ C of i(P …”

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confidence: 99%

Roots of Ehrhart polynomials of Gorenstein Fano polytopes

Hibi

Higashitani

Ohsugi

2011

Proc. Amer. Math. Soc.

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Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

Ohsugi

Tsuchiya

2020

Sel. Math. New Ser.

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In this paper, we introduce polytopes $${\mathscr {B}}_G$$ B G arising from root systems $$B_n$$ B n and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that $${\mathscr {B}}_G$$ B G is reflexive if and only if G is bipartite. Moreover, in the case, $${\mathscr {B}}_G$$ B G has a regular unimodular triangulation. This implies that the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the $$\gamma $$ γ -positivity and the real-rootedness of the $$h^*$$ h ∗ -polynomials. In fact, if G is bipartite, then the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is $$\gamma $$ γ -positive and its $$\gamma $$ γ -polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $$h^*$$ h ∗ -polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose $$h^*$$ h ∗ -polynomial is not real-rooted but $$\gamma $$ γ -positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

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Roots of Ehrhart polynomials arising from graphs

Cited by 48 publications

References 20 publications

Interlacing Ehrhart polynomials of reflexive polytopes

Interlacing Ehrhart polynomials of reflexive polytopes

Roots of Ehrhart polynomials of Gorenstein Fano polytopes

Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

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