The question was cross-posted from Math.SE: <https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck>




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The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\colon\mathbb C^n\to\mathbb C^n$ that is injective is also surjective.



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> Is assuming $p$ has finite fibers enough?
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I can't come up with any easy counter-examples, but the proof I know using finite fields does not work.



Strengthening Ax-Grothendieck
A counterexample for $n = 2$ is the map $(x\_1, x\_2) \mapsto (x\_1x\_2 - 1, x\_2(x\_1x\_2 - 1) + x\_1)$. This example, which I learned from a paper of Zbigniew Jelonek, was mentioned in an [earlier answer of mine](https://mathoverflow.net/a/343994/1508).


