# Bernstein-Sato polynomials for general ideals vs. principal ideals

## TL;DR

This paper establishes a relationship between Bernstein-Sato polynomials of ideals and principal ideals, and explores implications for the Strong Monodromy Conjecture in the context of Igusa zeta functions.

## Contribution

It demonstrates that the Bernstein-Sato polynomial of an ideal equals the reduced polynomial of a related function, linking invariants of ideals to those of principal functions.

## Key findings

- Bernstein-Sato polynomial of an ideal equals that of a related principal function.
- Relates invariants of arbitrary ideals to principal ideals via a constructed function.
- Shows that the Strong Monodromy Conjecture for principal ideals implies it for all ideals.

## Abstract

We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with an r-dimensional affine space. By combining this with results from [BMS], we relate invariants and properties of I to those of g. We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.