Bernstein–Sato root criterion for general homogeneous polynomials

Establish that if a homogeneous polynomial f of degree d in n variables satisfies, for every nontrivial decomposition of the dual vector space into complementary subspaces (C^n)^*=W_1\oplus W_2 and every integer k with 0≤k≤d and nk≠d·dim W_1, the condition that f does not lie in the image of Sym^k(W_1)⊗Sym^{d-k}(W_2)→Sym^d((C^n)^*), then -n/d is a root of the local Bernstein–Sato polynomial b_{f,0}(s).

Background

The paper gives examples of indecomposable homogeneous polynomials for which -n/d is not a root of the Bernstein–Sato polynomial, showing that indecomposability alone does not suffice for arbitrary homogeneous polynomials. These examples motivate a more refined condition based on how the polynomial decomposes relative to direct-sum decompositions of the dual vector space.

The proposed conjecture asserts that excluding all such weighted decompositions should force -n/d to be a root of the local Bernstein–Sato polynomial. The subsequent remark explains that polynomials lying in one of the excluded images yield a vanishing top de Rham cohomology class, providing the motivation for the conjectural sufficient condition.

References

Furthermore, we propose the following conjecture. Let $ f \in Sym{d}((C{n}){*}) $ be a homogeneous polynomial of degree $ d $ with $ n $ variables. If for any nontrivial decomposition $ (C{n}){*} = W_{1}\oplus W_{2} $ and any integer $ 0 \leq k \leq d $ such that $ nk \neq d \cdot \dim W_{1} $, we all have that $ f $ is not contained in the image of $ Sym{k}(W_{1}) \otimes Sym{d-k}(W_{2}) \rightarrow Sym{d}((C{n}){*}) $, then $ -\frac{n}{d} $ is a root of the local Bernstein--Sato polynomial $ b_{f,0}(s) $.

— The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements  (2501.05189 - Xie et al., 9 Jan 2025) in Section 5, Examples of homogeneous polynomials with b_f(-n/d)≠0; Conjecture 5.1