Counting polynomials for linear codes, hyperplane arrangements, and matroids

Abstract

Thomas-decomposition of a polynomial systems and the resulting counting polynomials are applied to the theory of linear codes, hyperplane arrangements, and vector matroids to reinterpret known polynomials such as characteristic polynomials and weight enu- merator, to introduce a new polynomial counting the matrices defin- ing the same matroid, and to introduce the concept of essential flats revealing a structure which allows to rewrite the rank generating poly- nomial as a sum of products of univariate polynomials. Our concepts make no essential distinction between finite and infinite fields.