Artin approximation theorem: Difference between revisions

Revision as of 03:33, 12 February 2012

Revision as of 19:51, 30 July 2012

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Let

:'''x''' = ''x''₁, …, ''x''_''n''

denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s,

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''k'''''<nowiki>[[x]]</nowiki>''' the [[ring (mathematics)|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 equation]]s in ''k''['''x''', '''y'''], and ''c'' a positive [[integer]]. Then given a formal power series solution '''ŷ'''('''x''') ∈ ''k'''''<nowiki>[[x]]</nowiki>''' there is an algebraic solution '''y'''('''x''') consisting of [[algebraic function]]s such that

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==References==

*{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebraic approximation of structures over complete local rings | url=http://www.numdam.org/item?id=PMIHES_1969__36__23_0 | idmr={{MR|0268188}} | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=36 | pages=23–58}}

*Artin, Michael. ''Algebraic Spaces''. Yale University Press, 1971.