On the total degree of certain L-functions

Assume a polynomial f is an element of F-q[x, y] and an additive character psi of F-q are given. From a set of exponential sums defined by f and psi o ne can define an L-function L-f(t), which by results of Dwork and Grothedie ck is known to be a rational function. In fact, L-f(t) is the Artin L-funct ion associated to psi and to an Artin-Schreier covering defined from f. In this note we give bounds for the number of poles of L-f(t) and for its tota l degree (the number of zeros plus the number of poles). Our bounds are giv en in terms of the Newton polyhedron of f. The bound for the total degree w e give improves, for polynomials in two variables, previous bounds of E. Bo mbieri (1978, Invent. Math. 47, 29-35) and A. Adolphson-S. Sperber (1987, I nvent. Math. 88, 555-569). (C) 2001 Academic Press.