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arXiv · 2512.20442

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

Yakob Kahane
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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.

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Yakob Kahane. 2025-12-23. A Proof of a conjecture of Watanabe--Yoshida via Ehrhart Theory. https://arxiv.org/abs/2512.20442

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