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

### Tristan Torrelli

Université Henri Poincaré, Vandoeuvre lès Nancy, France

## Abstract

Let *f*, *g* be two germs of holomorphic functions on 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.