About: Ax–Grothendieck theorem

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In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck. The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field.

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rdf:type	yago:WikicatTheoremsInAlgebra yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Ax–Grothendieck theorem (en) Théorème d'Ax-Grothendieck (fr) アックス–グロタンディークの定理 (ja)
rdfs:comment	In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck. The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field. (en) En mathématiques, le théorème d'Ax-Grothendieck est un résultat d'algèbre sur l'injectivité et la surjectivité des polynômes qui a été prouvé indépendamment par James Ax et Alexandre Grothendieck. Ce théorème est souvent énoncé dans le cas particulier suivant : toute fonction polynomiale de Cn dans Cn qui est injective est bijective. Le théorème complet est la généralisation à n'importe quelle variété algébrique sur un corps algébriquement clos. (fr) 数学において、アックス–グロタンディークの定理（英: Ax–Grothendieck theorem）は単項式の単射性と全射性についての結果である。 (James Ax) とアレクサンドル・グロタンディーク (Alexander Grothendieck) によって独立に証明された。定理はしばしば次の特別な形で述べられる。P が Cn から Cn への多項式関数で、P が単射ならば、P は全単射である。つまり、P が異なる引数を常に異なる値に写すならば、P の値域は Cn 全体を覆う。定理の完全な形は代数的閉体上の任意の代数多様体に一般化される。 (ja)
dcterms:subject	Theorems in algebra Model theory
Wikipage page ID	21867395 (xsd:integer)
Wikipage revision ID	1062073689 (xsd:integer)
Link from a Wikipage to another Wikipage	Model theory Morphism of finite type Algebraic closure Algebraically closed field Homomorphism Analytic function Radicial morphism Bijective Compactness theorem Garden of Eden (cellular automaton) Alexander Grothendieck Algebraic variety Amenable group Finite field Cellular automaton Hilbert's Nullstellensatz James Ax Armand Borel Theorems in algebra Model theory Characteristic (algebra) Polynomial Picard theorem Injective Éléments de géométrie algébrique Injectivity Complex vector space Surjectivity
Link from a Wikipage to an external page	http://xorshammer.wordpress.com/2008/08/15/axs-theorem/%7Ctitle=Ax%E2%80%99s
sameAs	Ax–Grothendieck theorem Ax–Grothendieck theorem Ax–Grothendieck theorem Ax–Grothendieck theorem Ax–Grothendieck theorem Ax–Grothendieck theorem Ax–Grothendieck theorem
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has abstract	In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck. The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field. (en) En mathématiques, le théorème d'Ax-Grothendieck est un résultat d'algèbre sur l'injectivité et la surjectivité des polynômes qui a été prouvé indépendamment par James Ax et Alexandre Grothendieck. Ce théorème est souvent énoncé dans le cas particulier suivant : toute fonction polynomiale de Cn dans Cn qui est injective est bijective. Le théorème complet est la généralisation à n'importe quelle variété algébrique sur un corps algébriquement clos. (fr) 数学において、アックス–グロタンディークの定理（英: Ax–Grothendieck theorem）は単項式の単射性と全射性についての結果である。 (James Ax) とアレクサンドル・グロタンディーク (Alexander Grothendieck) によって独立に証明された。定理はしばしば次の特別な形で述べられる。P が Cn から Cn への多項式関数で、P が単射ならば、P は全単射である。つまり、P が異なる引数を常に異なる値に写すならば、P の値域は Cn 全体を覆う。定理の完全な形は代数的閉体上の任意の代数多様体に一般化される。 (ja)
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foaf:isPrimaryTopicOf	wikipedia-en:Ax–Grothendieck_theorem
is Link from a Wikipage to another Wikipage of	List of mathematical logic topics Compactness theorem Garden of Eden (cellular automaton) List of New York University alumni James Ax Bijection Ax-Grothendieck theorem List of theorems List of things named after Alexander Grothendieck Surjunctive group
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