2016/11/08 by Eduard Yu. Lerner, Lerner, Eduard Yu.
Mathematics · #05B35 #05C31 #11T06 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05B35 #msc:05C31 #msc:11T06
paper · pdf · doi:10.48550/arxiv.1611.02746
17 pages
arxiv created 2016/11/08 · arxiv updated 2016/11/10
Let M=(E,\mathcal B) be an \mathbb Fq-linear matroid; denote by \mathcal B the family of its bases, s(M;α)=∑B∈\mathcal B∏e ∈ B αe, where αe∈ \mathbb Fq. According to the Kontsevich conjecture stated in 1997, the number of nonzero values of s(M;α) 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^⊥ in terms of the "correct" Kontsevich formula (for \mathbb Fq-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^⊥ in two ways, namely, in terms of characteristic polynomials of M/A and M|A, respectively, A⊆ E. The latter expressions are close to convolution-multiplication formulas established by V. Reiner and J. P. S. Kung.