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Roots of Ehrhart polynomials arising from graphs
 J. ALGEBR. COMB
, 2010
"... 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 ..."
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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 + d4  ≤ d4 or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip −D2 ≤ Re(α) ≤ D2 −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.
GALE DUALITY BOUNDS FOR ROOTS OF POLYNOMIALS WITH NONNEGATIVE COEFFICIENTS
, 2007
"... We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analy ..."
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We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.
Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part (Extended Abstract)
, 2012
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ASYMPTOTIC DISTRIBUTION OF THE ROOTS OF THE EHRHART POLYNOMIAL OF THE CROSSPOLYTOPE
, 2010
"... We use the method of steepest descents to study the root distribution of the Ehrhart polynomial of the ddimensional crosspolytope, namely Ld, as d → ∞. We prove that the distribution function of the roots, approximately, as d grows, by variation of argument of the generating function m≥0 Ld(m)tm ..."
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We use the method of steepest descents to study the root distribution of the Ehrhart polynomial of the ddimensional crosspolytope, namely Ld, as d → ∞. We prove that the distribution function of the roots, approximately, as d grows, by variation of argument of the generating function m≥0 Ld(m)tm+x−1 = (1 + t)d(1 − t)−d−1tx−1, as t varies appropriately on the segment of the imaginary line contained inside the unit disk.