Bernstein--Sato Polynomials for Positively Weighted Homogeneous Locally Everywhere Divisors, Hyperplane Arrangements, in C^3

We consider the Bernstein--Sato polynomial of a polynomial f R = Cx₁, x₂, x₃ that analytically locally everywhere admits a positively weighted homogeneous defining equation. We construct, in the analytic category, a complex of Dₗs-modules that can be used to compute the Dₗs-dual of Dₗs f^s-1 as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narv\'aez 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₌^0 (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\'aez 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 C^3 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₌^0 (R / (f) ) ₃₄₆ (₅) - ₁.