The $\alpha$-representation for the Tait coloring and for the characteristic polynomial of matroid

Abstract

Consider a finite field $\mathbb F_q$, $q=p^d$, where $p$ is an odd number. 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 the maximal 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 the linear combination of Legendre symbols with respect to $\bar s(M|A;\alpha)$, whose coefficients are modulo equal to $1/q^{r^*(\alpha)/2}$. This representation generalizes the formula for a 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 in the case of a finite field, which has inspired the Kontsevich conjecture (1997). The $\alpha$-representation technique is also applicable for expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the matrix of faces of this graph.