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

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