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Deformations of polynomials and their zeta functions

S. M. Gusein-Zade, D. Siersma

Abstract

For an analytic family P_s of polynomials in n variables (depending on a complex number s, and defined in a neighborhood of s = 0), there is defined a monodromy transformation h of the zero level set V_s= {P_s=0} for s different from 0, small enough. The zeta function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta functions. We show some examples of computations with the use of this technique.

Deformations of polynomials and their zeta functions

Abstract

For an analytic family P_s of polynomials in n variables (depending on a complex number s, and defined in a neighborhood of s = 0), there is defined a monodromy transformation h of the zero level set V_s= {P_s=0} for s different from 0, small enough. The zeta function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta functions. We show some examples of computations with the use of this technique.

Paper Structure

This paper contains 4 sections, 2 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The localization principle
  3. Localization for families of polynomials
  4. Examples

Key Result

Theorem 1

One has and therefore

Figures (1)

  • Figure 1: The Newton diagrams of the summands corresponding to deformation at the point $(0:0:0:1)$ (marked by $\bullet$ and $\circ$ respectively).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2