Polynomials Counting Nowhere-Zero Chains Associated with Homomorphisms.

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from Z E orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let A ψ [ N ] be the set of vectors g from (A − 0) E , such that ∑ e ∈ E g (e) · f (e) = ψ (f) for each f ∈ N (where · is a scalar multiplication). We show that | A ψ [ N ] | can be evaluated by a polynomial function of | A | . In particular, if ψ (f) = 0 for each f ∈ N , then the corresponding assigning polynomial is the classical characteristic polynomial of M. [ABSTRACT FROM AUTHOR]