Articles tagged with "model-theory"

Ax-Grothendieck Theorem

\(\newcommand{\CC}{\Bbb C} \newcommand{\FF}{\Bbb F} \newcommand{\QQ}{\Bbb Q} \newcommand{\FFx}[1]{\overline{\FF_{#1}}} \newcommand{\ACF}{\mathbf{ACF}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cT}{\mathcal{T}}\)

The Ax-Grothendieck theorem is the statement:

Ax-Grothendieck Theorem

Let \(f: \CC^n \to \CC^n\) be a polynomial map; that is, each coordinate \(f_i: \CC^n \to \CC\) is a polynomial in the \(n\) input variables. Then, if \(f\) is injective, it is surjective.

This… doesn’t seem like a particularly exciting theorem. But it has a really exciting proof.