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 ); and an algebraic version of this theorem in 1969.

Statement of the theorem

Let denote a collection of n indeterminates, k\mathbf{x} the ring of formal power series with indeterminates \mathbf{x} over a field k, and a different set of indeterminates. Let be a system of polynomial equations in, and c a positive integer. Then given a formal power series solution, there is an algebraic solution 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 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 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, there is a such that