Bounds of characteristic polynomials of regular matroids

A regular chain group $N$ is the set of integral vectors orthogonal to rows of a matrix representing a regular matroid, i.e., a totally unimodular matrix. Introducing canonical forms of an equivalence relation generated by $N$ and a special basis of $N$, we improve several results about polynomials counting elements of $N$ and find new bounds and formulas for these polynomials

Large Circuits in Binary Matroids of Large Cogirth: I

Let F 7 denote the Fano matroid and e be a fixed element of F 7 . Let P (F 7 ; e) be the family of matroids obtained by taking the parallel connection of one or more copies of F 7 about e. Let M be a simple binary matroid such that every cocircuit of M has size at least d 3. We show that if M does not have an F 7 -minor, M 6= F 7 and d (r(M) + 1)=2 then M has a circuit of size r(M) + 1. We also show that if M is connected, e 2 E(M ), M does not have both an F 7 -minor and an F 7 -minor, and M = 2 P (F 7 ; e), then M has a circuit that contains e and has size at least d + 1