Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gröbner methods

Basile Coron

Abstract

Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done using a quadratic Gröbner basis associated to a new presentation of those rings, which is obtained by iterating the semi-small decomposition of Chow rings of matroids. This Gröbner basis can also be applied to compute certain principal ideals in these rings, and ultimately reestablish the known connection between derangement polynomials and the Hilbert series of Chow rings for corank 1 uniform matroids. More broadly, this approach enables us to express the Hilbert series of Chow rings for any uniform matroid as polynomials related to the ascent statistics on particular sets of inversion sequences.

An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gröbner methods

Abstract

Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done using a quadratic Gröbner basis associated to a new presentation of those rings, which is obtained by iterating the semi-small decomposition of Chow rings of matroids. This Gröbner basis can also be applied to compute certain principal ideals in these rings, and ultimately reestablish the known connection between derangement polynomials and the Hilbert series of Chow rings for corank 1 uniform matroids. More broadly, this approach enables us to express the Hilbert series of Chow rings for any uniform matroid as polynomials related to the ascent statistics on particular sets of inversion sequences.

Paper Structure

This paper contains 3 sections, 16 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Chow rings of boolean matroids and Eulerian polynomials
  3. Chow rings of uniform matroids and derangement polynomials

Key Result

Proposition 2.3

For all loopless matroid $\mathrm{M}$ on a finite set $E$, the Chow ring $\underline{\mathrm{CH}}(\mathrm{M})$ is isomorphic to the quotient algebra where $\mathcal{I}$ is the ideal generated by the elements $h_{\overline{\imath}} \, (i \in E)$ with $\overline{\imath}$ the intersection of the flats containing $i$, and $\mathcal{J}$ is the ideal generated by the elements $(h_F - h_{F_1})(h_F - h_{

Theorems & Definitions (38)

  • Definition 2.1: Matroid
  • Definition 2.2: Chow ring of a matroid
  • Proposition 2.3: Pagaria_2023 Theorem 2.9
  • Definition 2.4: Eulerian polynomial
  • Proposition 2.5: Braden_2022 Remark 2.11
  • Proposition 2.6: BW_1993 Theorem 5.35
  • Definition 2.7: Initial segment
  • Proposition 2.9: coron_2023 Proposition 3.2, Theorem 3.6
  • Example 2.10
  • Proposition 2.11
  • ...and 28 more