On the Tensor Property of Bernstein-Sato Polynomial

TL;DR

This paper proves a multiplicative property of Bernstein-Sato polynomials for divisors and extends it to ideals, providing new insights into their tensor behavior and connections to monodromy conjectures.

Contribution

It establishes a multiplicative Thom-Sebastiani rule for Bernstein-Sato polynomials and extends the result to tensor products of divisors and monomial ideals.

Findings

Proved the Thom-Sebastiani rule for Bernstein-Sato polynomials.

Extended the rule to tensor products of divisors on complex varieties.

Proposed and proved an extension to Bernstein-Sato polynomials for monomial ideals.

Abstract

We prove the multiplicative Thom-Sebastiani rule for Bernstein-Sato polynomials, answering the longstanding questions of Budur and Popa. We generalize the result to the tensor of two effective divisors on the product of two arbitrary non-singular complex varieties. This also leads to a multiplicative property related to Igusa's strong monodromy conjecture. Moreover, we propose an extension of our result to Bernstein-Sato polynomials for ideals and prove it for monomial ideals.