T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

We show that every quasi-ordinary Weierstrass polynomial $P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] $, $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero $\K$, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant $\Delta_P \in \K[[X]]$ is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of $\K[[X]]$ and the function germs of quasi-analytic families

SUR LA LINÉARITÉ DE LA FONCTION DE ARTIN PAR GUILLAUME ROND RÉSUMÉ.-Nous donnons ici un contre-exemple à une vieille conjecture en théorie des singularités. Cette conjecture est que la fonction qui apparaît dans la version forte du théorème d'approximation de Artin est majorée par une fonction affine. Tout d'abord nous faisons une étude de l'approximation diophantienne entre le corps des séries en plusieurs variables et son complété pour la topologie m-adique. Nous montrons, à l'aide d'un exemple, qu'il n'existe pas de version du théorème de Liouville dans ce contexte. Ce même exemple nous fournit notre contre-exemple (théorème 1.2). Nous appliquons cela pour donner une nouvelle preuve du fait qu'il n'existe pas de théorie de l'élimination des quantificateurs pour le corps des séries en plusieurs variables.  2005 Elsevier SAS ABSTRACT.-We give here a counterexample to an old conjecture in the theory of singularities. This conjecture is that the function that appears in the strong Artin approximation theorem is bounded by an affine function. First we study Diophantine approximation between the field of power series in several variables and its completion for the m-adic topology. We show, with an example, that there is no Liouville theorem in this case. This example gives us our counterexample (cf. théorème 1.2). As an application, we give a new proof of the fact that there is no theory of elimination of quantifiers for the field of fractions of the ring of power series in several variables.

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 1 X 2 − X 3 X 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.