On the Total Degree of Certain L-Functions

Abstract

Assume a polynomial f∈Fq[x, y] and an additive character ψ of Fq are given. From a set of exponential sums defined by f and ψ one can define an L-function Lf(t), which by results of Dwork and Grothedieck is known to be a rational function. In fact, Lf(t) is the Artin L-function associated to ψ and to an Artin–Schreier covering defined from f. In this note we give bounds for the number of poles of Lf(t) and for its total degree (the number of zeros plus the number of poles). Our bounds are given in terms of the Newton polyhedron of f. The bound for the total degree we give improves, for polynomials in two variables, previous bounds of E. Bombieri (1978, Invent. Math.47, 29–39) and A. Adolphson–S. Sperber (1987, Invent. Math.88, 555–569).