The Ax-Grothendieck Theorem states that any polynomial map from complex n-dimensional space to itself that is injective must also be surjective. Named for James Ax and Alexander Grothendieck, its best known proof uses methods from mathematical logic, reducing the statement over the complex numbers to the case of finite fields, where an injective map on a finite set is automatically surjective, a striking case of model-theoretic technique settling a purely algebraic-geometric question.
Facts
StatementIf a polynomial map from an n-dimensional complex vector space to itself sends distinct points to distinct points, then its values cover the whole space, so the map is automatically a bijection. The result generalizes to regular maps on any algebraic variety over an algebraically closed field. 1 Proof YearGrothendieck's proof, cited here, appeared in 1966; the same References list separately dates James Ax's independent proof to 1968. 1966 is recorded here as the earlier of the two independent dates. Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Ax-Grothendieck Theorem (Wikipedia)
Proved By
Sources
1. Ax-Grothendieck Theorem (Wikipedia)
Wikimedia FoundationLead section, the special-case statement
If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective.
References
Grothendieck, A. (1966). Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. Vol. 28. pp. 103-104, Theorem 10.4.11.
In Branch: Algebra, Lead
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.
In Group: Polynomials and Elementary Algebra, Lead section
theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.