Transcendence of the Artin-Mazur Zeta Function for Polynomial Maps of A^1(F_p)

Abstract

We study the rationality of the Artin-Mazur zeta function of a dynamical system defined by a polynomial self-map of A^1(k), where k is the algebraic closure of the finite field F_p. The zeta functions of the maps f(x)=x^m for (p,m)=1 and f(x)=x^{p^m}+ax for nonzero a in F_{p^m}, p odd, are shown to be transcendental.