Ehrhart polynomial roots of reflexive polytopes

Akihiro Higashitani; Alexander Kasprzyk; G\'abor Heged\"us

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arxiv 1503.05739 v1 pith:NYITUIAH submitted 2015-03-19 math.CO math.AG

Ehrhart polynomial roots of reflexive polytopes

G\'abor Heged\"us , Akihiro Higashitani , Alexander Kasprzyk This is my paper

classification math.CO math.AG

keywords polytopepolytopesreflexiverootsconditionehrhartexamplehalf-strip

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Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.

Ehrhart polynomial roots of reflexive polytopes

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