Lemme dʹArtin–Rees, théorème dʹIzumi et fonction de Artin

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 X1X2−X3X4, 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.