Roots of Ehrhart polynomials arising from graphs
Roots of Ehrhart polynomials arising from graphs
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由图产生的埃尔哈特多项式的根
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
T. Hibi
中科院分区：
文献类型：
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作者：
Tetsushi Matsui;A. Higashitani;Yuuki Nagazawa;Hidefumi Ohsugi;T. Hibi
有几个多面体是由有限图产生的。特别是，对于边和对称边多面体，Ehrhart多项式的穷举计算不仅支持Beck等人的猜想。证明了D维多面体的Ehrhart多项式的所有根α满足−D≤Re(α)≤D−1)，但也揭示了每类多面体的一些有趣的现象。本文提出了两个新的猜想：(1)对于d阶完全多部图，边多面体的Ehrhart多项式的根位于圆$|z+Frac{d}{4}|le Frc{d}{4}$或为负整数；(2)D的Gorenstein Fano多面体的根在更窄的条带$-Frac{D}{2}leq maththm{Re}(α)leq Frac{D}{2}-1$中。得到了支持它们和原猜想的一些严谨的结果。每种多面体的Ehrhart多项式的根分布如图所示。
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle $|z+frac{d}{4}| le frac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip $-frac{D}{2} leq mathrm{Re}(alpha) leq frac{D}{2}-1$. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.