A note on the one-dimensional systems of formal equations
type: Item
instance of: scholarly article
title: A note on the one-dimensional systems of formal equations
zbMATH Open document ID: 0705.14003
author: J. Elias
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74843
review text: Let \(F_ i=0\), \(i=1,...,s\), \(f_ i\in kX_ 1,...,X_ N\) a system of formal equations over an infinite field k and assume that it defines a (nonnecessarily reduced) algebroid curve \(\gamma\). The author gives, from Milnor's \(\mu\) of \(\gamma_{red}\), an explicit determination of a function \(\beta\) (n) with the following property: for any r, \(1\leq r\leq N\) and any set of power series \(G_ i(X_ 1,...,X_ r)\in kX_ 1,...,X_ r\), \(i=r+1,...,N\) giving an approximate solution of the equations modulo \((X_ 1,...,X_ r)^{\beta (n)}\), this is, \(F_ i(X_ 1,...,X_ r,G_{r+1},...,G_ n)\equiv 0,\) \(i=1,...,s,\) modulo \((X_ 1,...,X_ r)^{\beta (n)}\), there are series \(\bar G_ i\in kX_ 1,...,X_ r\), \(i=r+1,...,N\) giving a true solution, i.e., \(F_ i(X_ 1,...,X_ r,\bar G_{r+1},...,\bar G_ N)=0,\) \(i=1,...,s,\) and such that \(\bar G_ i\equiv G_ i\quad mod\quad (X_ 1,...,X_ r)^ n,\) for \(i=r+1,...,N.\) The existence of the function \(\beta\) was already established by \textit{M. Artin} [Publ. Math., Inst. Hautes Étud. Sci. 36(1969), 23-58 (1970; Zbl 0181.488)] with no hypothesis about the dimension of the solutions \(\gamma\). An effective determination of \(\beta\) for the case of \(\gamma\) being a reduced plane curve is due to \textit{J. Wavrick} [Trans. Am. Math. Soc. 245, 409-417 (1978; Zbl 0403.13009)].
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13B40
zbMATH DE Number: 4156611
zbMATH Keywords: Artin approximation
zbMATH Keywords: system of formal equations
zbMATH Keywords: algebroid curve
reviewed by: Eduardo Casas-Alvero
MaRDI profile type: Publication
OpenAlex ID: W2318645830
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DOI: 10.5802/AIF.1179
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 4156611