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We study the rationality of the Artin-Mazur zeta function of a dynamicalsystem defined by a polynomial self-map of A^1(k), where k is the algebraicclosure 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 betranscendental.

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