Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k.

Statement of the theorem

Let

x = x₁, …, x_n

denote a collection of n indeterminates,

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

y = y₁, …, y_m

a different set of indeterminates. Let

f(x, y) = 0

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 such that

ŷ(x) ≡ y(x) mod (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 moduli spaces from knowledge of the existence of certain formal moduli spaces of deformations as schemes.

References

• Artin, Michael. Algebraic Spaces. Yale University Press, 1971.