  Given $p$ polynomials with coefficients in a commutative unitary integral
ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic
Bernstein-Sato polynomial on an irreducible affine scheme $V \subset
\text{Spec}(\mathcal{C})$. We prove the existence of such a non zero rational
polynomial which covers and generalizes previous existing results by H. Biosca.
When $\mathcal{C}$ is the ring of an algebraic or analytic space, we deduce a
stratification of the space of the parameters such that on each stratum, there
is a non zero rational polynomial which is a Bernstein-Sato polynomial for any
point of the stratum. This generalizes a result of A. Leykin obtained in the
case $p=1$.
