Ax-Grothendieck Theorem

Theorem
Let $f : \C^n \to \C^n$ be a polynomial map.

If $f$ is injective, then $f$ is surjective.

Proof
The proof is done by


 * 1) showing that the theorem can be captured using a first-order sentences in the language of rings,
 * 2) showing that the theorem is true for at least one field of every characteristic $p>0$, and
 * 3) applying the Lefschetz Principle (First-Order).


 * Expressing the theorem using first-order sentences in the language of rings

Since $n$ is fixed, we can quantify over polynomials where each variable occurs with degree at most $d$ since there are only finitely many coefficients to quantify over. Thus, for each $n$ and $d$, we can build an $\mathcal{L}_r$ sentence which holds in a field $F$ if and only if every injective polynomial map $F^n \to F^n$ where each variable occurs with at most degree $d$ is surjective.

First, we write a formula $\phi_{(i_1,\dots,i_n)}$ which says that the $n$-variable polynomial map with coefficients $a_{(i_1,\dots,i_n)}$ where $(i_1,\dots,i_n)\leq (d,\dots,d)$ is injective. Note that the polynomial map takes on $n$-tuple images, so is of the form $f = (f_1, \dots, f_n)$ where each $f_k$ is a polynomial in $n$ variables. The variables $a_{k,(i_1,\dots,i_n)}$ in the sentence below are intended to be interpreted as the coefficients of $f_k$.
 * $ \displaystyle \forall x_1 \cdots \forall x_n \forall y_1 \cdots \forall y_n \left( \left( \bigwedge_{k\leq n}\ \sum_{(i_1,\dots,i_n)} a_{k,(i_1,\dots,i_n)} x_{1}^{i_1}\cdots  x_{n}^{i_n} = \sum_{(i_1,\dots,i_n)} a_{k,(i_1,\dots,i_n)}  y_{1}^{i_1}\cdots y_{n}^{i_n} \right) \rightarrow  \bigwedge_{i=1,\dots,n} x_i = y_i \right) $

We also write a formula $\psi_{(i_1,\dots,i_n)}$ which says that such a polynomial is surjective:
 * $ \displaystyle \forall z_1 \cdots \forall z_n \exists x_1 \cdots \exists x_n \left(\bigwedge_{k\leq n}\ \sum_{(i_1,\dots,i_n)} a_{k,(i_1,\dots,i_n)}  x_{1}^{i_1}\cdots x_{n}^{i_n} = z_k \right) $

Finally, we combine these into the required implication, quantifying over all coefficients.
 * $\displaystyle \underset{k,(i_1,\dots,i_n)}{\huge{\forall}} a_{k,(i_1,\dots,i_n)} \left[ \phi_{(i_1,\dots,i_n)} \rightarrow \psi_{(i_1,\dots,i_n)} \right]$

Note that we have one of these sentences for every maximum degree $d$ of the variables in a polynomial map.


 * Showing the theorem is true for at least one field of every characteristic $p>0$

Since injections on finite sets are necessarily surjective, every injective polynomial map $k^n \to k^n$ is surjective when $k$ is a finite field. We extend this to the algebraic closure of $k$. This will demonstrate that the sentence above is satisfied by at least one model of the theory of algebraically closed fields of characteristic $p$ for every $p>0$.

Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements, and suppose there is an injective polynomial map $f:(\mathbb{F}^{\operatorname{alg}}_p )^n \to (\mathbb{F}^{\operatorname{alg}}_p )^n$ which is not surjective.

Let $A$ be the set of coefficients appearing in $f$, and let $(z_1,\dots,z_n)\in (\mathbb{F}^{\operatorname{alg}}_p )^n$ be an element not in the range of $f$.

Consider the subfield $k$ of $\mathbb{F}^{\operatorname{alg}}_p$ generated by the elements of $A$ and the elements $z_1,\dots,z_n$. Since $\mathbb{F}^{\operatorname{alg}}_p = \displaystyle{\bigcup_{n=1,2,\dots} \mathbb{F}_{p^n}}$, any finitely generated subfield is contained in some finite sub-union $\displaystyle{\bigcup_{n=1,2,\dots,N} \mathbb{F}_{p^n}}$, and hence $k$ is finite.

Therefore, $f\restriction_{k^n}$ is an injective polynomial map on a finite field which is not surjective. This is a contradiction. So, we must have that every injective polynomial map on $(\mathbb{F}^{\operatorname{alg}}_p )^n$ is surjective. That is, $(\mathbb{F}^{\operatorname{alg}}_p )^n$ satisfies the sentences above for each characteristic $p>0$.


 * Transfer to $\C^n$.

By the Lefschetz Principle (First-Order), since we have shown that the sentences above are true in some algebraically closed field of characteristic $p$ for all $p>0$, they are true in $\C$.