  In this paper we provide a geometric description of the possible poles of the
Igusa local zeta function associated to an analytic mapping and a locally
constant function, in terms of a log-principalizaton of an ideal naturally
attached to the mapping. Typically our new method provides a much shorter list
of possible poles compared with the previous methods. We determine the largest
real part of the poles of the Igusa zeta function, and then as a corollary, we
obtain an asymptotic estimation for the number of solutions of an arbitrary
system of polynomial congruences in terms of the log-canonical threshold of the
subscheme given by the ideal attached to the mapping. We associate to an
analytic mapping a Newton polyhedron and a new notion of non-degeneracy with
respect to it. By constructing a log-principalization, we give an explicit list
for the possible poles of the Igusa zeta function associated to a
non-degenerate mapping.
