On Bernstein-Sato polynomials

Abstract:

We show that for fixed $n$ and $d$ the set of Bernstein-Sato polynomials of all the polynomials in at most $n$ variables of degrees at most $d$ is finite. As a corollary, we show that there exists an integer $t$ depending only on $n$ and $d$ such that $f^{-t}$ generates $R_f$ as a module over the ring of the $k$-linear differential operators of $R$, where $k$ is an arbitrary field of characteristic 0, $R$ is the ring of polynomials in $n$ variables over $k$ and $f\in R$ is an arbitrary non-zero polynomial of degree at most $d$.