Artin approximation theorem
- 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".
tatement of the theorem
Let
:x = "x"1, …, "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 = "y"1, …, "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 certain formal moduli spaces of deformations as schemes.
References
*Artin, Michael. "Algebraic Spaces". Yale University Press, 1971.
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