Some Results on Formal Power Series and Differentiable Functions

In [2] we see that any formal power series in two variables with coefficients in R or C (in this paper only the real case will be considered,) can be transformed to a polynomial by some automorphism change of the variables. In [3] Whitney shows an example which is a convergent series in three variables but which cannot be transformed to a polynomial. In this paper we give a formal power series example in three variables that is never transformed to be convergent (§ 2). A formal power series is the Taylor expansion of some C°° function at the origin by E. Bore] theorem. The followings refine it.