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==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 [[scheme (mathematics)|scheme]]s. See also: [[Artin's criterion]].
==Alternative statement==
The following alternative statement is given in Theorem 1.12 of {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=1969}}.
Let ''R'' be a field or an excellent discrete valuation ring, let ''A'' be the henselization of an ''R''-algebra of finite type at a prime ideal, let ''m'' be a proper ideal of ''A'', let <math> \hat{A}</math> be the ''m''-adic completion of ''A'', and let
:''F'': (''A''-algebras) → (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 <math> \overline{\xi} \in F(\hat{A})</math> there is a <math> \xi \in F(A)</math> such that
:<math>\overline{\xi}</math> ≡ <math>\xi</math> mod ''m''<sup>''c''</sup>.
== See also ==
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