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

Yakob Kahane

TL;DR

The paper provides an explicit coefficient-by-coefficient formula for the shifted Ehrhart polynomial $\Omega(Z_n;t-1/2)$ of zigzag posets, recasting the WY conjecture in purely combinatorial terms. It introduces a decomposition-based framework that expresses each coefficient as a Hadamard-product-style combination of Euler-number data and linear coefficients of convex subposets, and then derives an explicit generating function for the shifted polynomials. The authors show that, for large enough $n$, the shifted Ehrhart values dominate the Euler-number lower bound, yielding an Ehrhart-theoretic proof of the Watanabe–Yoshida conjecture. The approach also extends to related posets (e.g., crown posets) and to other shifts, highlighting a versatile combinatorial methodology connecting poset polytopes with Hilbert–Kunz 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.

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

TL;DR

The paper provides an explicit coefficient-by-coefficient formula for the shifted Ehrhart polynomial of zigzag posets, recasting the WY conjecture in purely combinatorial terms. It introduces a decomposition-based framework that expresses each coefficient as a Hadamard-product-style combination of Euler-number data and linear coefficients of convex subposets, and then derives an explicit generating function for the shifted polynomials. The authors show that, for large enough , the shifted Ehrhart values dominate the Euler-number lower bound, yielding an Ehrhart-theoretic proof of the Watanabe–Yoshida conjecture. The approach also extends to related posets (e.g., crown posets) and to other shifts, highlighting a versatile combinatorial methodology connecting poset polytopes with Hilbert–Kunz 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 . 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.
Paper Structure (25 sections, 49 theorems, 101 equations, 4 figures, 1 table)

This paper contains 25 sections, 49 theorems, 101 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Notation
  4. Linear extension, Ideal
  5. Order Polynomial and Ehrhart polynomial
  6. Zigzag posets and Fibonacci Polytope
  7. Euler numbers
  8. Watanabe--Yoshida conjecture
  9. Decomposition formula
  10. The weighted-sum formula
  11. Generating function
  12. Constant and linear coefficients
  13. Homogenization, decomposition into elementary blocks
  14. Computation of the weights w(mu) for elementary decompositions
  15. Multiplicative behavior
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $P$ be a poset. Then where $\mathsf{Decompo}(P)$ denotes the set of all decompositions of $P$ into $k$ convex subposets, $e(\mu)$ is the number of ways to linearly order the blocks of the decomposition, and $c_1(P_i)$ is the linear coefficient of the order polynomial of $P_i$.

Figures (4)

  • Figure 1: A decomposition $\mu$ and the induced poset.
  • Figure 2: Illustration of the function $T$. In numerals, the ordered standard decomposition. And with letters, the order of the merges (alphabetical order). For example, if you apply $T$ twice, you will obtain $(\{5\}, \{2\}, \{ 1\}, \{6, 4\} , \{3\})$ and then $(\{5, 2\}, \{ 1\}, \{6, 4\} , \{3\})$.
  • Figure 3: Example of an elementary decomposition of size $3$. All the blocks except one have size one.
  • Figure 4: The $3$ ways to decompose $Z_6$ into $2$ subposets. The decomposition have respective weights $w_-(3)w_{+}(3)$, $w_-(5)$ and $U_5$.

Theorems & Definitions (95)

  • Theorem 1.1: shareshian2003newapproachorderpolynomials
  • Theorem 1.2: pak2025hilbertkunzmultiplicityquadricsehrhart
  • Conjecture 1.3: watanabe2005hilbert
  • Theorem 1.4: (Theorem \ref{['thm:3']})
  • Theorem 1.5
  • Corollary 1.6: Corollary \ref{['corro:finalinequality']}
  • Theorem 1.7: Theorem \ref{['thm:proof 3rd part WS conj']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 85 more