# Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (1969) 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 $\displaystyle{ k = \Complex }$); and an algebraic version of this theorem in 1969.

## Statement of the theorem

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

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

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

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

## 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 Michael Artin (1969).

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

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

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

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