On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities

Given two holomorphic functions f and g defined in two respective germs of complex analytic manifolds ( X , x ) and ( Y , y ), we know thanks to M. Saito that, as long as one of them is Euler homogeneous, the reduced (or microlocal) Bernstein-Sato polynomial of the Thom-Sebastiani sum f + g can be expressed in terms of those of f and g . In this note we give a purely algebraic proof of a similar relation between the whole functional equations that can be applied to any setting (not necessarily analytic) in which Bernstein-Sato polynomials can be defined.