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An injective polynomial map $p:\mathbb{C}^n\mapsto\mathbb{C}^n$ is surjective (Ax-Grothendieck theorem). What is known about the reverse implication (surjective implies injective)? Why does the model-theoretic proof of A-G not work in the other direction?

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What is known is that it is false: Let $f(z)=z^2$.

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As André Nicolas has pointed out, the answer to the first question is trivial. But I think the second question is a good one.

The model-theoretic proof of Ax-Grothendieck goes like this:

  1. Show that it suffices to prove the statement for algebrically closed fields of characteristic $p>0$.
  2. Reduce to a finite field over which a given polynomial (and the element we want to hit to show surjectivity) is defined.
  3. Prove the statement for all finite fields.

Steps 1 and 3 work fine. But Step 2 is a problem: Just because our map is surjective on an algebraically closed field of characteristic $p$ doesn't mean it's surjective when restricted to a finite subfield.

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It fails at the reduction to $\overline{\mathbb{F}_p}$. Namely, in the classic proof we say that by compactness and the completeness of $\mathsf{ACF}_p$ it suffices to prove the claim for a map $f:\overline{\mathbb{F}_p}\to\overline{\mathbb{F}_p}$. We then said that if $b$ weren't hit, then our map would restrict to an injective but non-surjective map $k\to k$ where $k$ is the field obtained by adjoining $b$ and the coefficients of our map to $\mathbb{F}_p$. But, this is ridiculous since $k$ is finite.

If you tried to imitate this proof in the surjective-implies-injective case, the argument breaks down. Whereas before the global map $f$ being injective implies the map $k\to k$ is injective, the map $f$ being surjective does not imply the map $k\to k$ must be as well.

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Ah, you just beat me to it! Our answers are the same, but I would say it's the reduction to finite fields that fails, not the reduction to $\overline{\mathbb{F}}_p$... – Alex Kruckman May 12 '14 at 8:32
@AlexKruckman This is a deep question of grave importance to model theory--where precisely does the proof fail. Maybe we should email Hrushovski :) – Alex Youcis May 12 '14 at 8:45
No need - I think he'll be at msri tomorrow :) – Alex Kruckman May 12 '14 at 8:47

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