We show that if $f\in \mathcal{O}_X(X)$ and $g\in \mathcal{O}_Y(Y)$ are nonzero regular functions on smooth complex algebraic varieties $X$ and $Y$, then the Bernstein-Sato polynomial of the product function $fg \in \mathcal{O}_{X\times Y}(X \times Y)$ is given by $b_{fg}(s)=b_f(s)b_g(s)$. This answers a question of Mihnea Popa from his course notes \cite{Pop21} on $D$-modules.
A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors
From Differential Values to Roots of the Bernstein-Sato Polynomial
Zeta functions and Bernstein-Sato polynomials for ideals in dimension two
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