Sur la linearite de la fonction de Artin

Abstract

We give here a counter-example to a conjecture of Spivakovsky. M. Spivakovsky conjectured that the function that appears in the strong Artin approximation theorem is bounded by a linear function. First we show that there is no Liouville theorem for the field of fractions of the ring of power series in several variables. We deduce from this example that there is no theory of elimination of quantifiers for the field of fractions of the ring of power series in several variables.

To appear in Ann. Sci. Ecole Norm. Sup., changes following referee's comments. Proofs have been improved and the counter-example can be read independantly from the first part