  We interpret the Artin-Rees lemma and the Izumi theorem in term of Artin
function and we obtain a stable version of the Artin-Rees lemma. We present
different applications of these interpretations. First we show that the Artin
function of $X_1X_2-X_3X_4$, as a polynomial in the ring of power series in
more than three variables, is not bounded by an affine function. Then we prove
that the Artin functions of a class of polynomials are bounded by affine
functions and we use this to compute approximated integral closures of ideals.
