Artin approximation theorem: Difference between revisions

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==Statement of the theorem==

Let <math>\mathbf{x} = x_1, \dots, x_n</math> denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s, <math>k[\![\mathbf{x}]\!]</math> the [[ring (mathematics)|ring]] of formal power series with indeterminates <math>\mathbf{x}</math> over a field ''k'', and <math>\mathbf{y} = y_1, \dots, y_n</math> a different set of indeterminates. Let

Let

:<math>f(\mathbf{x}, \mathbf{y}) = 0</math>

:'''x''' = ''x''₁, …, ''x''_''n''

be a system of [[polynomial equation]]s in ''<math>k''['''\mathbf{x'''}, '''\mathbf{y'''}]</math>, 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 (more precisely, algebraic power series) such that

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

''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 (more precisely, algebraic power series) such that

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