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Sur le théorème de Maillet. (On Maillet’s theorem). (French) Zbl 0693.34004
For $$s\geq 0$$ denote by $${\mathbb{C}}\{x\}_ s$$ the space of series $$\sum^{\infty}_{0}a_ px^ p$$, $$a_ p\in {\mathbb{C}}$$, such that $$\sum (a_ p/(p!)^ s)x^ p$$ converges in a neighbourhood of 0. Let $$n\geq 0$$ be a fixed integer and F a holomorphic function of $$(n+2)$$ variables $$(x,y_ 0,...,y_ n)$$ in a neighbourhood of $$(0,a_ 0,...,a_ n)$$. For $$\phi\in {\mathbb{C}}[[x]]$$ with $$\partial^ k\phi(0)=a_ k$$ ($$0\leq k\leq n$$, $$\partial =d/dx)$$ set $$F(x,\phi)=F(x,\phi,\partial \phi,...,\partial^ n\phi).$$ If F is not identically 0, then if $$F(x,\phi)=0$$, there exists $$s\geq 0$$, such that $$\phi \in {\mathbb{C}}\{x\}_ s$$. Under the additional hypothesis that $$\partial F/\partial y_ n(x,\phi)$$ is not identically 0, the author gives an explicit estimate for s in terms of a certain Newton polygon.
Reviewer: I.N.Baker

##### MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 32A20 Meromorphic functions of several complex variables
Newton polygon