Bernstein Polynomials of a Smooth Function Restricted to an Isolated Hypersurface Singularity

Abstract

Let $f,g$ be two germs of holomorphic functions on ${\mathbf{C}}^n$ such that $f$ is smooth at the origin and $(f,g)$ defines an analytic complete intersection $(Z,0)$ of codimension two. We study Bernstein polynomials of $f$ associated with sections of the local cohomology module with support in $X=g^{-1}(0)$, and in particular some sections of its minimal extension. When $(X,0)$ and $(Z,0)$ have an isolated singularity, this may be reduced to the study of a minimal polynomial of an endomorphism on a finite dimensional vector space. As an application, we give an effective algorithm to compute those Bernstein polynomials when $f$ is a coordinate and $g$ is non-degenerate with respect to its Newton boundary.