Ax–Grothendieck theorem

An injective polynomial self-map of affine space over an algebraically closed field is surjective. The theorem holds in every finite dimension and in every characteristic.

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Polynomial map: A map whose coordinate functions are polynomials. The theorem concerns polynomial maps from affine space to itself.
James Ax: A mathematician known for work in model theory, number theory, and algebraic geometry. Ax established the theorem through model-theoretic methods.
Polynomial automorphism: A polynomial map with a polynomial inverse. The theorem says an injective polynomial self-map is bijective, though not by itself that its inverse is polynomial.
Injective non-surjective functions: Functions that send distinct inputs to distinct outputs while omitting at least one codomain element. Such functions exist on infinite sets, but not as polynomial self-maps under the theorem's hypotheses.
Affine space: The geometric space with coordinates over a field, modeled algebraically by tuples of field elements. It is the domain and codomain in the theorem.
Alexander Grothendieck: A mathematician whose work reshaped algebraic geometry through schemes and cohomological methods. Grothendieck gave an independent route to the same injectivity-surjectivity result.
Jacobian conjecture: The conjecture that a polynomial map with constant nonzero Jacobian determinant has a polynomial inverse. It concerns when local nonsingularity forces global invertibility, a nearby but stronger issue.
Polynomial maps over the real numbers: Maps between real affine spaces whose coordinate functions are real polynomials. Real polynomial self-maps can be injective without being surjective, showing the field condition matters.
Algebraically closed field: A field in which every nonconstant polynomial in one variable has a root. The theorem's surjectivity conclusion depends on this hypothesis.
Model theory: The mathematical study of structures through formal languages and their theories. Ax's proof uses logical transfer between algebraically closed fields.