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

Let

denote a collection of n indeterminates,

k[[x]] the ring of formal power series with indeterminates x over a field k, and

a different set of indeterminates. Let

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

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.

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