Bernstein polynomials of a smooth function restricted to an isolated hypersurface singularity
type: Item
instance of: scholarly article
title: Bernstein polynomials of a smooth function restricted to an isolated hypersurface singularity
zbMATH Open document ID: 1064.32022
author: Tristan Torrelli
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-11-01T00:00:00Z
review text: Set \(\mathcal{O}=\mathbb{C} \{x_1,x_2,\ldots,x_n\}\) the ring of germs at \(0\) of complex holomorphic functions, \(g\in \mathcal{O}\), such that \(g(0)=0\), and \(\mathcal{R}= \mathcal{O}[1/g] / \mathcal{O}\) the local cohomology module with support in the hypersurface \((X,0)\) defined by \(g\). For a germ of function \(f \in \mathcal{O}\) nonzero on \(X\), there are functional equations in \( \mathcal{R}\otimes \mathcal{O}[1/f,s] f^s\) of the form \(b(s)\delta f^s=P \cdot \delta f^{s+1}\). The unitary generator of the ideal of polynomials \(b(s)\) is called to be the Bernstein polynomial of \(f\) associated with \(\delta\). It is well known that roots of these polynomials determine the eigenvalues of the monodromy of the restriction \(f| _X :(X,0) \rightarrow (\mathbb{C},0)\). The author studies these polynomials in the particular case where \(f\) is a germ of smooth function and gives an algorithm to compute them when \(f\) is a coordinate and \(g\) in non-degenerate with respect to its Newton boundary.
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 2113015
zbMATH Keywords: Bernstein polynomial
zbMATH Keywords: isolated surface singularity
reviewed by: Carlos Galindo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145476048
OpenAlex ID: W1986446282
cites work: Le réseau \(L^ 2\) d'un système holonome régulier. (The \(L^ 2\)- réseau of a regular holonomic system)
cites work: Algorithme de calcul du polynome de Bernstein. Cas non dégéneré. (Algorithm of computation of Bernstein polynomials. The non-degenerate case)
cites work: Q4064285
cites work: Characteristic varieties and vanishing cycles
cites work: On the De Rham cohomology of algebraic varieties
cites work: On the holonomic systems of linear differential equations. II
cites work: Q3681128
cites work: Q4746079
cites work: Q3872745
cites work: Q4065784
cites work: Polynômes de Bernstein-Sato associés à une intersection complète quasi-homogène à singularité isolée
cites work: Local cohomology of analytic spaces
cites work: Q3759138
cites work: Q3769182
cites work: Divisions by the fractional powers of an ideal generated by a regular sequence in the ring of germs of analytic functions at the origin of \(\mathbb{C}^2\)
cites work: Équations fonctionnelles pour une fonction sur une intersection complète quasi homogène à singularité isolée
cites work: Polynômes de Bernstein associés à une fonction sur une intersection complète à singularité isolée. (Bernstein polynomials associated with a function on an isolated complete intersection singularity)
DOI: 10.2977/PRIMS/1145476048
description: scientific article; zbMATH DE number 2113015