Asymptotic Distribution Of The Roots Of The Ehrhart Polynomial Of The Cross-Polytope

Abstract

We use the method of steepest descents to study the root distribution of the Ehrhart polynomial of the $d$-dimensional cross-polytope, namely ${\mathcal{L}}_d$, as $d\to \infty$. We prove that the distribution function of the roots, approximately, as $d$ grows, by variation of argument of the generating function ${\sum }_{m\ge 0}{\mathcal{L}}_d(m)t^{m+x-1}=(1+t{)}^d(1-t{)}^{-d-1}{t}^{x-1}$, as $t$ varies appropriately on the segment of the imaginary line contained inside the unit disk.

References in corpus (1)

• Notes on the roots of Ehrhart polynomials