Let X be a X, and denote by F=f1…fp their product. Given a regular holonomic DX-module M and a section m∈M, denote by B(x,f1,…,fp,m) the Bernstein–Sato ideal of C[s1,…,sp] consisting of polynomials b(s1,…,sp) such that there exists, in a neighborhood of x∈F−1(0), a differential operator P(s1,…,sp)∈DX⊗CC[s1,…,sp] satisfying P(s1,…,sp)mfs1+11…fsp+1p=b(s1,…,sp)mfs11…fspp. Claude Sabbah proved that this ideal is nonzero. One can associate to the characteristic variety of the DX[s1,…,sp]-module DX[s1,…,sp]mfs11…fspp a finite set Hf,m of hyperplanes in Cp. We prove that there exists a Bernstein–Sato polynomial (i.e., a nonzero member of the Bernstein–Sato ideal) which is a product of one variable polynomials if and only if the set Hf,m is contained in the union of the coordinate hyperplanes. In the two variables case (p=2) we prove that there exist a Bernstein–Sato polynomial the higher degree form of which vanishes on and only on the set Hf,m.