Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field "k" are well-approximated by the algebraic functions on "k".
More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case formula_1); and an algebraic version of this theorem in 1969.
Statement of the theorem.
Let formula_2 denote a collection of "n" indeterminates, formula_3 the ring of formal power series with indeterminates formula_4 over a field "k", and formula_5 a different set of indeterminates. Let 
be a system of polynomial equations in formula_7, and "c" a positive integer. Then given a formal power series solution formula_8, there is an algebraic solution formula_9 consisting of algebraic functions (more precisely, algebraic power series) such that 
Discussion.
Given any desired positive integer "c", this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by "c". This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes. See also: Artin's criterion.
Alternative statement.
The following alternative statement is given in Theorem 1.12 of .
Let formula_11 be a field or an excellent discrete valuation ring, let formula_12 be the henselization of an formula_11-algebra of finite type at a prime ideal, let "m" be a proper ideal of formula_12, let formula_15 be the "m"-adic completion of formula_12, and let 
be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then for any integer "c" and any formula_18, there is a formula_19 such that 