Ax-Grothendieck theorem

proper noun

A result about injectivity and surjectivity of polynomials, 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. The full theorem generalizes to any algebraic variety over an algebraically closed field.

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Meanings

proper nounthe Ax-Grothendieck theorem
  1. mathematicsA result about injectivity and surjectivity of polynomials, 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. The full theorem generalizes to any algebraic variety over an algebraically closed field.

Origin

Proved independently by James Ax and Alexander Grothendieck.

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Cite this entry
APA

Ax-Grothendieck theorem. (n.d.). In WatkinsX Dictionary. Retrieved November 4, 2026, from https://dictionary.watkinsx.com/word/Ax-Grothendieck%20theorem

MLA

“Ax-Grothendieck theorem.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/Ax-Grothendieck%20theorem. Accessed November 4, 2026.

Chicago

WatkinsX Dictionary, s.v. “Ax-Grothendieck theorem,” accessed November 4, 2026, https://dictionary.watkinsx.com/word/Ax-Grothendieck%20theorem.

Source: Ax-Grothendieck theorem on Wiktionary, CC BY-SA 4.0.