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- W1820287328 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 $F$ nâest 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 ${mathcal {A}}circ F$." @default.
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- W1820287328 date "2001-01-02" @default.
- W1820287328 modified "2023-09-26" @default.
- W1820287328 title "On composite formal power series" @default.
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- W1820287328 doi "https://doi.org/10.1090/s0002-9947-01-02733-7" @default.
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