  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.

Commutative Algebra