A note on the one-dimensional systems of formal equations
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)].
scientific article; zbMATH DE number 4156611
scientific article