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 ${\displaystyle P}$ is an injective polynomial function from an ${\displaystyle n}$-dimensional complex vector space to itself then ${\displaystyle P}$ is bijective. That is, if ${\displaystyle P}$ always maps distinct arguments to distinct values, then the values of ${\displaystyle P}$ cover all of ${\displaystyle \mathbb {C} ^{n}}$.

The full theorem generalizes to any algebraic variety over an algebraically closed field.