Artin-Rees lemma and the Izumi theorem in term of Artin function

摘要

We interpret the Artin-Rees lemma and the Izumi theorem in term of Arlin function and we obtain a stable version of the Artin-Rees lemma. We present different applications of these interpretations. First we show that the Arlin 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 Arlin functions of a class of polynomials are bounded by affine functions and we use this to compute approximated integral closures of ideals. (c) 2005 Elsevier Inc. All rights reserved.