Bernstein--Sato Polynomials for Positively Weighted Homogeneous Locally Everywhere Divisors, Hyperplane Arrangements, in C³

We consider the Bernstein--Sato polynomial of a polynomial f ∈ R = C[x₁, x₂, x₃] that analytically locally everywhere admits a positively weighted homogeneous defining equation. 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 Narv\'{a}ez 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 Hₘ⁰ (R / (∂ f)) contributes a root to the Bernstein--Sato polynomial, generalizing a result of Saito's (where the argument cannot weaken homogeneity to positive homogeneity). Second, we prove the zeroes of Bernstein--Sato polynomial admit a partial symmetry about $-1$, extending a result of Narv\'{a}ez 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 Hₘ⁰( R / (∂ f))deg(f) - 1.