The $\alpha$-Representation for Tait Coloring and the Characteristic Polynomial of a Matroid

Abstract

Consider a finite field $\mathbb F_q$, $q=p^d$, where $p$ is an odd prime. Let $M=(E,r)$ be a regular matroid; denote by ${\mathcal B}$ the family of its bases, $\bar s(M;\alpha)=\sum_{B\in{\mathcal B}}\prod_{e\not\in B} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$, $\alpha_e\neq 0$. Let a subset $A\equiv A(\alpha)$ in $E$ have maximum cardinality and satisfy the condition $\bar s(M|A;\alpha)\neq 0$, while $r^*(\alpha)=|A|-r(E)$. Let us represent the value of the characteristic polynomial of the matroid $M$ at the point $q$ as a linear combination of Legendre symbols with respect to $\bar s(M|A;\alpha)$, whose coefficients are equal in modulus to $1/q^{r^*(\alpha)/2}$. This representation generalizes the formula for the flow polynomial of a graph which was obtained by us earlier. The latter formula is an analog of the so-called $\alpha$-representation of vacuum Feynman amplitudes over finite fields, which inspired the Kontsevich conjecture (1997). The $\alpha$-representation technique is also applicable to expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the face matrix of this graph.