The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables

TL;DR

This paper proves that a broad class of homogeneous polynomials in three variables satisfies the Strong Monodromy Conjecture, linking roots of Bernstein--Sato polynomials to monodromy in a motivic framework.

Contribution

It characterizes when  is a root of the Bernstein--Sato polynomial for a class of polynomials and proves the Strong Monodromy Conjecture for these in three variables.

Findings

Characterization of roots of Bernstein--Sato polynomial for the class of polynomials.

Proof that these polynomials satisfy the Strong Monodromy Conjecture in three variables.

Extension of the conjecture to a broad class of homogeneous polynomials with isolated singularities.

Abstract

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.