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 $$k = \Complex$$); and an algebraic version of this theorem in 1969.

Statement of the theorem
Let $$\mathbf{x} = x_1, \dots, x_n$$ denote a collection of n indeterminates, $$k\mathbf{x}$$  the ring of formal power series with indeterminates $$\mathbf{x}$$ over a field k, and $$\mathbf{y} = y_1, \dots,  y_n$$ a different set of indeterminates. Let


 * $$f(\mathbf{x}, \mathbf{y}) = 0$$

be a system of polynomial equations in $$k[\mathbf{x}, \mathbf{y}]$$, and c a positive integer. Then given a formal power series solution $$\hat{\mathbf{y}}(\mathbf{x}) \in k\mathbf{x}$$, there is an algebraic solution $$\mathbf{y}(\mathbf{x})$$ consisting of algebraic functions (more precisely, algebraic power series) such that


 * $$\hat{\mathbf{y}}(\mathbf{x}) \equiv \mathbf{y}(\mathbf{x}) \bmod (\mathbf{x})^c.$$

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 $$R$$ be a field or an excellent discrete valuation ring, let $$A$$ be the henselization at a prime ideal of an $$R$$-algebra of finite type, let m be a proper ideal of $$A$$, let $$\hat{A}$$ be the m-adic completion of $$A$$, and let


 * $$F\colon (A\text{-algebras}) \to (\text{sets}),$$

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 $$ \overline{\xi} \in F(\hat{A})$$, there is a $$ \xi \in F(A)$$ such that


 * $$\overline{\xi} \equiv \xi \bmod m^c$$.