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
Let
x=x1,...,xn
k[[x]]
x
y=y1,...,yn
f(x,y)=0
be a system of polynomial equations in
k[x,y]
\hat{y
y(x)
\hat{y
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.
The following alternative statement is given in Theorem 1.12 of .
Let
R
A
R
A
\hat{A}
A
F\colon(A-algebras)\to(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}\inF(\hat{A})
\xi\inF(A)
\overline{\xi}\equiv\xi\bmodmc