Mathematical Conversations

The Ax–Grothendieck theorem from the 1960s says that an injective polynomial $f \colon \mathbb{C}^n \rightarrow \mathbb{C}^n$ is also surjective. It is one of the first examples of the powerful technique in algebraic geometry of using finite fields to prove results over the complex numbers. We will give an outline of Ax's proof, which is virtually devoid of algebro-geometric input.