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

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. U. 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. We also give some essential indecomposable homogeneous polynomials $f$ such that $-{n\over d}$ is not a root of $b_{f,0}$. This leads to a conjectural sufficient condition for $b_{f,0}(-{n\over d})=0$.