2024/02/13 by Daniel Bath, Bath, Daniel
Mathematics · #Advanced Algebra and Geometry #Holomorphic and Operator Theory #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2402.08342
We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial f ∈ R = ℂ[x1, x2, x3]. We construct, in the analytic category, a complex of \mathscrDX[s]-modules that can be used to compute the \mathscrDX[s]-dual of \mathscrDX[s] fs-1 as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeroes of the Bernstein--Sato polynomial. First, we prove each nonvanishing degree of the zeroeth local cohomology of the Milnor algebra H_\mathfrakm0 (R / (∂ f)) contributes a root to the Bernstein--Sato polynomial, generalizing a result of M. Saito's (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeroes of the Bernstein--Sato polynomial admit a partial symmetry about -1, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted Logarithmic Comparison Theorem, and more precise statements when f is additionally assumed to be homogeneous. Finally, when f defines a hyperplane arrangement in ℂ3 we give a complete formula for the zeroes of the Bernstein--Sato polynomial of f. We show all zeroes except the candidate root -2 + (2 / deg(f)) are (easily) combinatorially given; we give many equivalent characterizations of when the only non-combinatorial candidate root -2 + (2/ deg(f)) is in fact a zero of the Bernstein--Sato polynomial. One equivalent condition is the nonvanishing of H_\mathfrakm0( R / (∂ f))deg(f) - 1.