Sur la linéarité de la fonction de Artin

We give here a counter-example 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 counter-example (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.