Rationality of partial zeta functions (Q1433053)

Let \(X\) be an algebraic variety defined over the finite field with \(q\) elements. In his previous work [\textit{D. Wan}, Finite Fields Sppl. 7, 238--251 (2001 Zbl 0995.14004)], the author has defined a system of partial zeta-functions associated to \(X\). He now proves that each of his zeta-functions is rational and that the absolute values of its poles and zeros are of the form \(q^{-w/2}\) for some nonnegative integer \(w\). This result is further strengthened in [\textit{L. Fu} and \textit{D. Wan}, Math. Ann. 328, 193--228 (2004; Zbl 1130.11033)]. As an application of his results, the author investigates certain zeta-functions of graphs and states an asymptotic formula for the number of rational points of an Artin-Schreier hypersurface, which has been proved in his joint paper with L. Fu cited above.