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

TL;DR

This paper analyzes the asymptotic distribution of roots of the Ehrhart polynomial for the cross-polytope in high dimensions using complex analysis techniques, revealing how roots distribute as dimension increases.

Contribution

It provides a novel asymptotic analysis of the roots of Ehrhart polynomials for cross-polytopes using the method of steepest descents and argument variation of the generating function.

Findings

Roots distribute along specific curves as dimension increases

Distribution function approximates variation of argument of the generating function

Method applies steepest descents to analyze root locations asymptotically

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 $d\rightarrow \infty$. We prove that the distribution function of the roots, approximately, as $d$ grows, by variation of argument of the generating function $\sum_{m\geq 0}\mathcal{L}_{d}(m)t^{m+x-1}=(1+t)^{d}(1-t)^{-d-1}t^{x-1}$, as $t$ varies appropriately on the segment of the imaginary line contained inside the unit disk.