The $\alpha$-representation for the characteristic function of a matroid

TL;DR

This paper introduces a new representation for the characteristic polynomial of a dual matroid using the Kontsevich formula, extending previous work on flow polynomials and connecting to convolution formulas.

Contribution

It expresses the dual matroid's characteristic polynomial via the Kontsevich formula for $ ext{F}_q$-linear matroids, generalizing flow polynomial formulas and linking to convolution-multiplication relations.

Findings

Derived a formula for the dual matroid's characteristic polynomial.

Connected the characteristic polynomial to flow and chromatic polynomials.

Extended the Kontsevich formula to $ ext{F}_q$-linear matroids.

Abstract

Let $M=(E,\mathcal B)$ be an $\mathbb F_q$-linear matroid; denote by ${\mathcal B}$ the family of its bases, $s(M;\alpha)=\sum_{B\in\mathcal B}\prod_{e \in B} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$. According to the Kontsevich conjecture stated in 1997, the number of nonzero values of $s(M;\alpha)$ is a polynomial with respect to $q$ for all matroids. This conjecture was disproved by P. Brosnan and P. Belkale. In this paper we express the characteristic polynomial of the dual matroid $M^\perp$ in terms of the "correct" Kontsevich formula (for $\mathbb F_q$-linear matroids). This representation generalizes the formula for a flow polynomial of a graph which was obtained by us earlier (and with the help of another technique). In addition, generalizing the correlation (announced by us earlier) that connects flow and chromatic polynomials, we define the characteristic polynomial of $M^\perp$ in two ways, namely, in terms of characteristic polynomials of $M/A$ and $M|_A$, respectively, $A\subseteq E$. The latter expressions are close to convolution-multiplication formulas established by V. Reiner and J. P. S. Kung.