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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 a Weierstrass-Hironaka division theorem for such subrings. Moreover, given an ideal ℐ of A and a series f in A we prove the existence in A of a unique remainder r modulo ℐ. As a consequence, we get a new proof of the noetherianity of A.

Let $F$ be a holomorphic map from ${ℂ}^s$ to ${ℂ}^s$ defined in a neighborhood of zero such that $F\left(0\right)=0.$ 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 $𝒜$ such that $𝒜\circ F$ 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...

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 ${ℂ}^s$ to ${ℂ}^p$ without constant term such that the rank of the Jacobian matrix of F is equal to 1. Let be a formal power series. If F is a holomorphic map, the following result is well known: ∘ F is analytic implies there exists a convergent power series...

We discuss some local analytic properties of the ring of Dirichlet series. We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one. In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series. Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.

Ce travail est une étude analytique locale de l’anneau des séries de Dirichlet convergentes. Dans un premier temps, on établit des propriétés arithmétiques de cet anneau ; on prouve en particulier sa factorialité, que l’on déduit de théorèmes de division du type Weierstrass. Ensuite, on s’intéresse à des problèmes de composition. Soient $f\left(s\right)$ et $\varphi \left(s\right)$ des séries de Dirichlet convergentes. On sait que $f(c_{0}s+\varphi \left(s\right)),$ avec $c_0\in {\mathbb{N}}^{*},$ est encore une série de Dirichlet convergente. On étudie la réciproque : sous les hypothèses que...

We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...