Sabbah defined a version of b-functions for multiple functions in his 1987 paper here: <http://archive.numdam.org/ARCHIVE/CM/CM_1987__62_3/CM_1987__62_3_283_0/CM_1987__62_3_283_0.pdf>


According to Wikipedia (<http://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato_polynomial>), this b-function factors into linear factors with rational coefficients. I don't see that in this paper; however, my French is terrible, and I probably have simply missed it. The claim itself seems to be unsourced, if it's not in Sabbah. 


Is this known true, and if it is, is there a proof of it in English?



Sabbah b-functions factoring
I emailed Dr. Sabbah, and he sent me a link to this paper by Gyoja: <http://projecteuclid.org/download/pdf_1/euclid.kjm/1250519266>


That paper proves the theorem on any affine variety. 


