Rationality criteria and application to the generating series of a system of equations with coefficients in a local field
type: Item
instance of: scholarly article
title: Rationality criteria and application to the generating series of a system of equations with coefficients in a local field
zbMATH Open document ID: 0606.12011
DOI: 10.1016/0022-314X(86)90031-4
author: Bernadette Deshommes
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: 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.
reviewed by: B. Z. Moroz
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 3979194
zbMATH Keywords: local zeta-functions
zbMATH Keywords: singularities
zbMATH Keywords: Poincaré series
zbMATH Keywords: poles
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3979194