On the Tensor Property of Bernstein-Sato Polynomial

M. Popa proposed a question whether bf· g(s) = bf(s)bg(s) holds for f∈ C[x₁,...,xₙ] and g∈ C[y₁,...,yₘ], where bf, bg, and bf· g are corresponding Bernstein-Sato polynomials. It is a subtle problem that seems clear but turns out highly non-trivial. In this paper, we give a positive answer for Popa's question and generalize the result to arbitrary non-singular complex varieties X and Y, and the tensor of effective divisors D₁ and D₂ on them respectively, and prove a generalized theorem. The theorem also leads to a multiplicative result of Igusa's strong monodromy conjecture. Moreover, we propose two analogous conjectures about Bernstein-Sato polynomials for ideals and prove them for monomial ideals.