# Anneaux de séries formelles à croissance contrôlée

Abstract : Given a sequence $M=\{M_n\}_{n\in\bkN}$ of real positive numbers
logarithmically convex, we study subrings $\Gamma_M$ of the ring
of formal power series in $s$ variables whose coefficients
satisfy growth conditions with respect to $M.$ Under very few restrictive
conditions on $M,$ we get in these rings composition theorems.
We study the following problem. Given $F$ in $(\Gamma_M)^{s},$ if
${\cal A}\circ F$ belongs to $\Gamma_M,$ which $\Gamma_N$ does the
series ${\cal A}$ belong to ?
We also prove that, given a good order on $\bkN^{s},$ we can divide any series
in $\Gamma_M$ by a finite family of series $f_1,\dots,f_p$ in such a way that
the quotients and the remainder belong to $\Gamma_M.$ Then we
discuss algebraic properties of $\Gamma_M$ as division properties
modulo an ideal, noetherianity and flatness. We get preparation theorems
of Malgrange type in these rings. We also prove a version of
Artin's theorem.
Keywords :
Document type :
Theses
Domain :

Cited literature [18 references]

https://tel.archives-ouvertes.fr/tel-00080323
Contributor : Augustin Mouze <>
Submitted on : Thursday, June 15, 2006 - 4:44:23 PM
Last modification on : Sunday, November 29, 2020 - 3:23:58 AM
Long-term archiving on: : Monday, April 5, 2010 - 10:54:27 PM

### Identifiers

• HAL Id : tel-00080323, version 1

### Citation

Augustin Mouze. Anneaux de séries formelles à croissance contrôlée. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2000. Français. ⟨tel-00080323⟩

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