The Igusa local zeta function associated with the singular cases of the determinant and the Pfaffian
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.
scientific article; zbMATH DE number 874203