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

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.