The n/d-conjecture for indecomposable central hyperplane arrangements

Prove that for every indecomposable central hyperplane arrangement defined by a homogeneous polynomial f of degree d in n variables over the complex numbers, the number -n/d is a root of the Bernstein–Sato polynomial b_f(s), thereby establishing the conjecture beyond the nonresonant cases treated in the paper.

Background

The paper studies when the rational number -n/d is a root of the Bernstein–Sato polynomial of a homogeneous polynomial of degree d in n variables. For essential indecomposable hyperplane arrangements, this question is known as the n/d-conjecture of Budur, Mustață, and Teitler, and its validity would imply the strong monodromy conjecture for hyperplane arrangements.

The main theorem proves the conjecture for weighted hyperplane arrangements satisfying an explicit nonresonance condition on all indecomposable subspaces. Because the result is conditional on that combinatorial condition, the conjecture remains unresolved in the general indecomposable central-arrangement setting.