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Abstract

We consider subrings $A$ of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps $F$ from ${{\mathbb C}}^{s}$ to ${{\mathbb C}}^{p}$ without constant term such that the rank of the Jacobian matrix of $F$ is equal to 1. Let ${\cal A}$ be a formal power series. If $F$ is a holomorphic map, the following result is well known: ${\cal A}\circ F$ is analytic implies there exists a convergent power series $\, \widetilde { \!{\cal A}\, }$ such that ${\cal A}\circ F= \, \widetilde {\! {\cal A}\, }\circ F.$ We get similar results when the map $F$ is no longer holomorphic.