  Given a family of analytic functions near 0 \in C^n parametrized by a smooth
space, we study the Bernstein polynomial of the fiber on an irreducible variety
V of the space of parameters and we show that it is generically constant. We
prove that this polynomial b(s) satisfies a functional equation on V from which
we derive a contructible stratification of the space of parameters by the
Bernstein polynomial of the fiber. When the hypersurface admits generically a
unique singularity at 0 \in C^n, we prove that b(s) is the generic Bernstein
polynomial in the sense of Brian\c{c}on-Geandier-Maisonobe. The tools for the
proofs are a formal generalization of an algorithm by Oaku and the generic
standard bases studied by the author.
