Rationality criteria and application to the generating series of a system of equations with coefficients in a local field (Q1084451)

Generalizing the construction of \textit{I. N. Bernstein} and \textit{S. I. Gel'fand} [Funkts. Anal. Prilozh. 3, No. 1, 84--85 (1969; Zbl 0208.15201)] the author introduces a zeta function of an algebraic variety defined over a finite extension of the field of \(p\)-adic numbers and proves that it is a rational function of several complex variables. As an application of this result, she obtains a new proof of rationality of the Poincaré series for any algebraic variety (cf. \textit{D. Meuser} [Math. Ann. 256, 303--310 (1981; Zbl 0471.12014)] and \textit{J. Denef} [Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)] for previous results in this direction) and describes a finite set containing each of the poles of this series.