A Proof of a conjecture of Watanabe--Yoshida via Ehrhart Theory

TL;DR

This paper proves a conjecture by Watanabe and Yoshida on the lower bound of Hilbert-Kunz multiplicity using Ehrhart theory, providing a combinatorial and algebraic approach that offers an alternative to previous analytic proofs.

Contribution

It introduces a new explicit combinatorial formula for the coefficients of the shifted Ehrhart polynomial related to the conjecture, enabling a different proof approach.

Findings

Established the conjecture using Ehrhart polynomial inequalities.

Derived a new explicit formula for Ehrhart polynomial coefficients.

Connected generating functions with Euler numbers and algebraic functions.

Abstract

In 2005, Watanabe and Yoshida formulated a conjecture for a lower bound of the Hilbert-Kunz multiplicity of local rings that was recently settled by Meng using analytic methods. More recently, Pak-Shapiro-Smirnov-Yoshida used Ehrhart theory to compute explicitly the multiplicity and reduced the conjecture to showing an inequality of the values of the Ehrhart polynomial of a zigzag poset shifted to $t - 1/2$. We completely realize their approach to give another proof of this Watanabe--Yoshida conjecture. The main ingredient of the proof relies on a new explicit combinatorial formula for the coefficients of this shifted Ehrhart polynomial. In terms of the generating function of the shifted polynomial, this formula manifests itself as a Hadamard product of the exponential generating function of Euler numbers and an explicit algebraic function.