- Artin approximation theorem
In

mathematics , the**Artin approximation theorem**is a fundamental result ofMichael Artin indeformation theory which implies thatformal power series with coefficients in a field "k" are well-approximated by thealgebraic function s on "k".**tatement of the theorem**Let

:

**x**= "x"_{1}, …, "x"_{"n"}denote a collection of "n" indeterminates,

"k"

the ring of formal power series with indeterminates**x****x**over a field "k", and:

**y**= "y"_{1}, …, "y"_{"m"}a different set of indeterminates. Let

:"f"(

**x**,**y**) = 0be a system of

polynomial equation s in "k" [**x**,**y**] , and "c" a positiveinteger . Then given a formal power series solution**ŷ**(**x**) ∈ "k" there is an algebraic solution**x****y**(**x**) consisting ofalgebraic function s 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 space s of deformations as schemes.**References***Artin, Michael. "Algebraic Spaces". Yale University Press, 1971.

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