The Igusa local zeta function associated with the singular cases of the determinant and the Pfaffian
type: Item
instance of: scholarly article
title: The Igusa local zeta function associated with the singular cases of the determinant and the Pfaffian
zbMATH Open document ID: 0851.11067
author: Margaret Robinson
published in: Journal of Number Theory
publication date: 1996-12-04T00:00:00Z
review text: In this paper, the author completes the determination of Igusa's local zeta function for the complex powers of determinant polynomials. Let \(K\) be a local field and let \(f(x)\) be a polynomial in \(n\)-variables with coefficients in the local field \(K\). We consider the integral \(Z(t) : = \int_{(K^n)^0} |f(x) |^s_K \;|dx |\), which is a rational function in \(t = q^{- s}\) \((s\) is a complex number and \(q\) is the cardinality of the residue field of \(K)\). Here, \(|\cdot |_K\) is an absolute value in \(K\); \(|dx |\) is the Haar measure on \(K^n\); \((K^n)^0\) is the set of integral points in \(K^n\). When \(f(x)\) is a determinant of a Hermitian matrix, Igusa had determined the explicit forms of \(Z(t)\) except for some cases. The author of this paper completes the calculation of \(Z(t)\) for determinants of Hermitian matrices by determining them for the singular cases of the determinants and the Pfaffians.
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 874203
zbMATH Keywords: Igusa local zeta function
zbMATH Keywords: determinant polynomials
zbMATH Keywords: Pfaffians
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: jnth.1996.0055
OpenAlex ID: W2059567720
DOI: 10.1006/JNTH.1996.0055
description: scientific article; zbMATH DE number 874203