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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On composite formal power series


Authors: Jacques Chaumat and Anne-Marie Chollet
Journal: Trans. Amer. Math. Soc. 353 (2001), 1691-1703
MSC (2000): Primary 13F25, 13J05, 32A05
Published electronically: January 2, 2001
MathSciNet review: 1806723
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Abstract: Let $F$ be a holomorphic map from ${\mathbb{C}}^{n}$ to ${\mathbb{C}}^{n}$ defined in a neighborhood of $0$ such that $F(0)=0$. If the Jacobian determinant of $F$ is not identically zero, P. M. Eakin et G. A. Harris proved the following result: any formal power series 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.

The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of ${\mathcal{A}}$ by the radius of convergence of ${\mathcal{A}}\circ F$.



RÉSUMÉ. Soit $F$ une application holomorphe de ${\mathbb{C}}^{n}$dans ${\mathbb{C}}^{n}$ définie dans un voisinage de $0$ et vérifiant $F(0)=0$. Si le jacobien de $F$ n'est pas identiquement nul au voisinage de $0$, P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle ${\mathcal{A}}$ telle que ${\mathcal{A}}\circ F$ est analytique est elle-même analytique. Si le jacobien de $F$ est identiquement nul, ils montrent que la conclusion précédente est fausse.

Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de ${\mathcal{A}}$ par celui de ${\mathcal{A}}\circ F$.


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Additional Information

Jacques Chaumat
Affiliation: U.M.R. C.N.R.S. 8628, Université Paris-Sud, Mathématiques - Bât. 425, 91405 Orsay Cedex, France
Email: jacques.chaumat@math.u-psud.fr

Anne-Marie Chollet
Affiliation: U.M.R. C.N.R.S. 8524, Université de Lille, U.F.R. de Mathématique,59655 Villeneuve D’Ascq Cedex, France
Email: chollet@aglae.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02733-7
PII: S 0002-9947(01)02733-7
Received by editor(s): December 12, 1997
Published electronically: January 2, 2001
Article copyright: © Copyright 2001 American Mathematical Society