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The Ax-Grothendieck result states that any polynomial injective function from $\mathbb{C}^n$ to itself is surjective. Is there such a statement for entire functions ?

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    $\begingroup$ I think Picard's theorem prevents such a result for $n=1$. $\endgroup$
    – Sylvain JULIEN
    Commented Oct 7, 2016 at 15:47
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    $\begingroup$ See math.stackexchange.com/questions/29758/entire-1-1-function for the case n=1. They are not only surjective, they are even linear. $\endgroup$
    – HeinrichD
    Commented Oct 7, 2016 at 15:53
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    $\begingroup$ @Julien: do you have an explicit counter example for $n=1$ ? $\endgroup$
    – Yoyo
    Commented Oct 7, 2016 at 15:53
  • $\begingroup$ @HeinrichD. Thx. $\endgroup$
    – Yoyo
    Commented Oct 7, 2016 at 15:58
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    $\begingroup$ @Yoyo .The answer is no.The key words are Fatou-Bieberbach domains. $\endgroup$
    – Mohan Ramachandran
    Commented Oct 7, 2016 at 16:20

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