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Let $M=(E,\mathcal B)$ be an $\mathbb F_q$-linear matroid; denote by${\mathcal B}$ the family of its bases, $s(M;\alpha)=\sum_{B\in\mathcalB}\prod_{e \in B} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$. According tothe Kontsevich conjecture stated in 1997, the number of nonzero values of$s(M;\alpha)$ is a polynomial with respect to $q$ for all matroids. Thisconjecture was disproved by P. Brosnan and P. Belkale. In this paper we expressthe characteristic polynomial of the dual matroid $M^\perp$ in terms of the"correct" Kontsevich formula (for $\mathbb F_q$-linear matroids). Thisrepresentation generalizes the formula for a flow polynomial of a graph whichwas obtained by us earlier (and with the help of another technique). Inaddition, generalizing the correlation (announced by us earlier) that connectsflow 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 toconvolution-multiplication formulas established by V. Reiner and J. P. S. Kung.

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