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In [[mathematics]], the '''Artin approximation theorem''' is a fundamental result of {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=1969}} in [[deformation theory]] which implies that [[formal power series]] with coefficients in a [[field (mathematics)|field]] ''k'' are well-approximated by the [[algebraic function]]s 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.
==Statement of the theorem==
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