Ax–Grothendieck theorem

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.