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, $\bar{s}(M;\alpha )={\sum }_{B\in \mathcal{B}}{\prod }_{e\notin B}{\alpha }_e$, where ${\alpha }_e\in {\mathbb{F}}_q$, ${\alpha }_e\ne 0$. Let a subset $A\equiv A(\alpha )$ in $E$ have the maximal cardinality and satisfy the condition $\bar{s}(M|A;\alpha )\ne 0$, while $r^{\ast }(\alpha )=|A|-r(E)$. 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 $1/q^{r^{\ast }(\alpha )/2}$. 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.

12 pages, 1 figure