Bernstein–Sato polynomials of locally quasi-homogeneous divisors in C³

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 DX[s] -modules that can be used to compute the DX[s] -dual of DX[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 zeros of the Bernstein–Sato polynomial. First, we prove each nonvanishing degree of the zeroth local cohomology of the Milnor algebra Hₘ⁰ (R / (∂ f)) contributes a root to the Bernstein–Sato polynomial, generalizing a result of M. Saito (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeros 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 zeros of the Bernstein–Sato polynomial of f . We show all zeros except the candidate root -2 + (2 / (f)) are (easily) combinatorially given; we give many equivalent characterizations of when the only noncombinatorial candidate root -2 + (2/ (f)) is in fact a zero of the Bernstein–Sato polynomial. One equivalent condition is the nonvanishing of Hₘ⁰( R / (∂ f))(f) - 1 .