Comments on: Infinite fields, finite fields, and the Ax-Grothendieck theorem

By: TH

TH — Tue, 23 Apr 2013 21:32:23 +0000

Hello there,
it seems to me that you have to consider polynomial maps (i.e. n-tuples of polynomials) instead of just polynomials, as you do. Then the equations coming from the Nullstellensatz have to be adapted accordingly (which is not hard), etc.
Best regards -

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By: Random

Random — Sun, 06 Mar 2011 03:49:36 +0000

Minor correction: in the proof of Corollary 5, z should be mapped to a random vector in C^n, not a random complex number.

[Corrected, thanks - T.]

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[...] Tao, Terence (2009-03-07). “Infinite fields, finite fields, and the Ax-Grothendieck theorem”. What’s New. [...]

By: Terence Tao

Terence Tao — Fri, 11 Dec 2009 21:15:09 +0000

Ah yes, that would work too :-)

By: L Spice

L Spice — Wed, 09 Dec 2009 20:23:34 +0000

I’m probably being silly, but I was confused by your proof that an injective polynomial self-map on the algebraic closure of a finite field is bijective. Rather than using the full power of the Nullstellensatz, why not just observe that is an injective polynomial map on all the finite fields containing its coefficients, hence is bijective on any such field, and then proceed by taking limits?

By: Terry Hughes

Terry Hughes — Mon, 04 May 2009 04:39:44 +0000

Terry,

If you like this theorem, perhaps you would be interested in some partial converses to it proved by Ken McKenna a while ago. Here are two:

1) A generically surjective polynomial map of n-dimensional affine space over a finitely generated extension of Z or Z/(p)[t] is bijective with a polynomial inverse raional over the same ring (and therefore bijective on affine space of the algebraic closure).

2) A generically surjective rational map of n-dimensional affine space over a Hilbertian field is generically bijective with a rational inverse defined over the same field.

Both proofs use model theory. McKenna used (1) to show some curious facts about the distribution of integral points of a hypothetical polynomial map defined over the integers with Jacobian equal to 1 that did not satisfy the Bieberbach conjecture. He never published, but some of the material was eventually included in a paper by Lou Van den Dries in Manuscripta Mathematica some years back.

By: Tricki now live « What’s new

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