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

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

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