  We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety
(which is not necessarily reduced nor irreducible), using the theory of
V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing
filtration by multiplier ideals coincides essentially with the restriction of
the V-filtration. This implies a relation between the roots of the
Bernstein-Sato polynomial and the jumping coefficients of the multiplier
ideals, and also a criterion for rational singularities in terms of the maximal
root of the polynomial in the case of a reduced complete intersection. These
are generalizations of the hypersurface case. We can calculate the polynomials
explicitly in the case of monomial ideals.
