The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements

TL;DR

This paper verifies the N/D-Conjecture for nonresonant hyperplane arrangements, linking Bernstein--Sato polynomials to topological monodromy, and confirms the conjecture for specific weighted arrangements using cohomological methods.

Contribution

It proves the N/D-Conjecture for a class of nonresonant hyperplane arrangements by applying Walther's cohomological criterion.

Findings

Confirmed the N/D-Conjecture for nonresonant arrangements

Connected the conjecture to topological monodromy implications

Utilized cohomological techniques to verify the conjecture

Abstract

This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements, this is a conjecture by Budur, Musta\c{t}\u{a} and Teitler and implies the strong topological monodromy conjecture for arrangements. Walther gave a sufficient condition that a certain differential form does not vanish in the top cohomology group of Milnor fiber. We use Walther's result to verify the $n\over d$-conjecture for weighted hyperplane arrangements satisfying the nonresonant condition.