  This paper is dedicated to the description of the poles of the Igusa local
zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy
condition, that we have called arithmetic non degeneracy. More precisely, we
attach to each polynomial $f(x,y)$, a collection of convex sets $\Gamma ^{A}$
called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of
arithmetic non degeneracy with respect to $\Gamma ^{A}(f)$. The set of
degenerate polynomials, with respect to a fixed geometric Newton polygon,
contains an open subset, for the Zariski topology, formed by non degenerate
polynomials with respect to some arithmetic Newton polygon.If $L$ is a number
field, our main result asserts that for almost all non-archimedean valuations
$v$ of $L$, the poles of $Z(s,f,v),$ with $f(x,y)\in L[x,y]$, can be described
explicitly in terms of the equations of the straight segments that conform the
boundaries of the convex sets that belong to $\Gamma ^{A}(f)$. Moreover, our
proof gives an effective procedure to compute $Z(s,f,v)$.
