Ax-Grothendieck Theorem

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Theorem

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

Let $f$ be injective.


Then $f$ is surjective.


Proof

The proof proceeds as follows:

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


First-Order Sentences in Language of Rings

Since $n$ is fixed, we can quantify over polynomials.

There are only finitely many coefficients to quantify over.

Therefore each variable occurs with degree at most $d$, say.

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:

$\phi_{\left({i_1, \dots, i_n}\right)}$

which says that the $n$-variable polynomial map with coefficients $a_{\left({i_1, \dots, i_n}\right)}$ where $(\left({i_1, \dots, i_n}\right) \le \left({d, \dots, d}\right)$ is injective.

Note that the polynomial map takes on $n$-tuple images, so is of the form:

$f = \left({f_1, \dots, f_n}\right)$

where each $f_k$ is a polynomial in $n$ variables.

The variables $a_{k, \left({i_1, \dots, i_n}\right)}$ in the sentence below are intended to be interpreted as the coefficients of $f_k$:

$\displaystyle \forall x_1 \cdots \forall x_n \forall y_1 \cdots \forall y_n \left({\left({\bigwedge_{k \mathop \le n} \ \sum_{\left({i_1, \ldots, i_n}\right)} a_{k, \left({i_1, \ldots, i_n}\right)} {x_1}^{i_1} \cdots {x_n}^{i_n} = \sum_{ \left({i_1, \ldots, i_n}\right)} a_{k, \left({i_1, \ldots, i_n}\right)} {y_1}^{i_1} \cdots {y_n}^{i_n} }\right) \to \bigwedge_{i \mathop = 1, \ldots, n} x_i = y_i}\right)$


We also write a formula $\psi_{(i_1,\dots,i_n)}$ which says that such a polynomial is surjective:

$\displaystyle \forall z_1 \cdots \forall z_n \exists x_1 \cdots \exists x_n \left(\bigwedge_{k \mathop \le n} \ \sum_{(i_1, \ldots, i_n)} a_{k, \left({i_1, \ldots, i_n}\right)} {x_1}^{i_1} \cdots {x_n}^{i_n} = z_k \right)$


Finally, we combine these into the required implication, quantifying over all coefficients:

$\displaystyle \underset{k, \left({i_1, \ldots, i_n}\right)} {\huge \forall} a_{k, \left({i_1, \ldots, i_n}\right)} \left({ \phi_{\left({i_1, \ldots,i_n}\right)} \to \psi_{\left({i_1, \ldots, i_n}\right)} }\right)$

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

Thus it has been demonstrated that the theorem can be captured using a first-order sentences in the language of rings.


$\Box$


Field of 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.

Suppose there is an injective polynomial map:

$f: \left({\mathbb F^{\operatorname{alg} }_p}\right)^n \to \left({\mathbb F^{\operatorname{alg} }_p}\right)^n$

which is not surjective.


Let $A$ be the set of coefficients appearing in $f$.

Let $\left({z_1, \ldots, z_n}\right) \in \left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ 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 $z_1, \ldots, z_n$.

We have that:

$\displaystyle \mathbb F^{\operatorname{alg} }_p = \bigcup_{n \mathop = 1, 2, \ldots} \mathbb F_{p^n}$

Therefore any finitely generated subfield is contained in some finite sub-union:

$\displaystyle \bigcup_{n \mathop = 1, 2, \ldots, N} \mathbb F_{p^n}$

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 it must be the case that every injective polynomial map on $\left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ is surjective.

That is:

$\left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ satisfies the sentences above for each characteristic $p > 0$.


Thus it has been demonstrated that the theorem is true for at least one field of every characteristic $p > 0$.

$\Box$


Application of Lefschetz Principle

It has been shown that the sentences above are true in some algebraically closed field of characteristic $p$ for all $p > 0$.

By the Lefschetz Principle, it follows that they are true in $\C$.

$\blacksquare$


Source of Name

This entry was named for James Burton Ax and Alexander Grothendieck.