Abstract

Let $F$ be a holomorphic map from $\mathbb{C}^{s}$ to $\mathbb{C}^{s}$ defined in a neighborhood of zero such that $F=0.$ If the jacobian determinant of $F$ is not identically zero, P. M. Eakin and G. A. Harris proved the following result: any formal power series ${\mathcal{A}}$ such that ${\mathcal{A}}\circ F$ is analytic is itself analytic. If the jacobian determinant of $F$ is identically zero, they proved that the previous conclusion is no more true. J. Chaumat and A.-M. Chollet extended this result in the case of formal power series satisfying growth conditions, of Gevrey type for instance. The author gets similar results when the map $F$ is no more holomorphic. The loss of regularity on ${\mathcal{A}}$ is optimal