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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

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