  We prove the multiplicative Thom-Sebastiani rule for Bernstein-Sato
polynomials, answering the longstanding questions of Budur and Popa. We
generalize the result to the tensor of two effective divisors on the product of
two arbitrary non-singular complex varieties. This also leads to a
multiplicative property related to Igusa's strong monodromy conjecture.
Moreover, we propose an extension of our result to Bernstein-Sato polynomials
for ideals and prove it for monomial ideals.
