Chapter

Finite Fields and Applications

Volume 2948 of the series Lecture Notes in Computer Science pp 109-136

Combinatorics of the Two-Variable Zeta Function

  • Iwan M. DuursmaAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign

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Abstract

We consider the rank polynomial of a matroid and some well-known applications to graphs and linear codes. We compare rank polynomials with two-variable zeta functions for algebraic curves. This leads us to normalize the rank polynomial and to extend it to a rational rank function. As applications to linear codes we mention: A formulation of Greene’s theorem similar to an identity for zeta functions of curves first found by Deninger, the definition of a class of generating functions for support weight enumerators, and a relation for algebraic-geometric codes between the matroid of a code and the two-variable zeta function of a curve.