Artin approximation theorem: Difference between revisions

Revision as of 15:08, 16 July 2009

Revision as of 20:35, 5 March 2010

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

''k''[['''[[x]]''']] the [[ring (mathematics)|ring]] of formal power series with indeterminates '''x''' over a field ''k'', and

: '''y''' = ''y''₁, …, ''y''_''m''

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:''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''[['''[[x]]''']] there is an algebraic solution '''y'''('''x''') consisting of [[algebraic function]]s such that

:'''ŷ'''('''x''') ≡ '''y'''('''x''') mod ('''x''')^''c''.