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Adding local short description: "Formal power series with coefficients in a field k are well-approximated by the algebraic functions on k", overriding Wikidata description "deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. Theorem as a fundamental result of Michael Artin" |
Citation bot (talk | contribs) Altered pages. Add: doi, volume. Formatted dashes. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Moduli theory | #UCB_Category 19/26 |
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{{Short description|
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''.
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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
:<math>f(\mathbf{x}, \mathbf{y}) = 0</math>
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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 | mr=0268188 | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=36 | issue=36 | pages=23–58| doi=10.1007/BF02684596 }}
*{{cite book|last=Artin|first= Michael|title=Algebraic Spaces|publisher= [[Yale University Press]]|series=Yale Mathematical Monographs|volume= 3|___location=New Haven, CT–London|year= 1971|mr=0407012}}
*{{citation|last=Raynaud|first= Michel|author-link=Michel Raynaud|title=Travaux récents de M. Artin| journal=[[Séminaire Nicolas Bourbaki]]|volume= 11 |year=1971|issue=363|pages=
[[Category:Moduli theory]]
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