Comments on: Infinite fields, finite fields, and the Ax-Grothendieck theorem
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoWed, 30 Jul 2014 00:20:03 +0000hourly1http://wordpress.com/By: Koushik Ghosh
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-246487
Thu, 26 Sep 2013 17:41:50 +0000http://terrytao.wordpress.com/?p=1869#comment-246487The jacobian conjecture gets reduced to asking injectivity of polynomial map from the rudin’s theorem if the jacobian is a nonzero constant.So,I am asking if the same holds if polynomials are replaced by holomorphic.
]]>By: Koushik Ghosh
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-246486
Thu, 26 Sep 2013 17:36:19 +0000http://terrytao.wordpress.com/?p=1869#comment-246486Is there some counterexample for the following “weaker jacobian conjecture for holomorphic functions” replace polynomials with holomorphic functions and instead of asking for invertibility asking just injectivity of the holomorphic map if the jacobian does not vanish or is a non-zero constant
]]>By: TH
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-225710
Tue, 23 Apr 2013 21:32:23 +0000http://terrytao.wordpress.com/?p=1869#comment-225710Hello 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 -
]]>By: The closed graph theorem in various categories « What’s new
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-194635
Wed, 21 Nov 2012 02:31:09 +0000http://terrytao.wordpress.com/?p=1869#comment-194635[...] several complex variables, it is a classical theorem (see e.g. Lemma 4 of this blog post) that a holomorphic function from a domain in to is locally injective if and only if it is a [...]
]]>By: Definable subsets over (nonstandard) finite fields, and almost quantifier elimination « What’s new
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-169851
Wed, 12 Sep 2012 23:36:40 +0000http://terrytao.wordpress.com/?p=1869#comment-169851[...] first studied systematically by Ax (in the same paper where the Ax-Grothendieck theorem, discussed previously on this blog, was established), with important further contributions by Kiefe, by Fried-Sacerdote by Cherlin-van [...]
]]>By: More On Coloring The Plane « Gödel’s Lost Letter and P=NP
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-52969
Sun, 22 May 2011 17:12:54 +0000http://terrytao.wordpress.com/?p=1869#comment-52969[...] A powerful meta-principle that I have discussed before is the close connection between questions about complex numbers and finite fields. For a much better explanation than I could give, please see the discussion a while ago by Terence Tao here. [...]
]]>By: Random
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-50592
Sun, 06 Mar 2011 03:49:36 +0000http://terrytao.wordpress.com/?p=1869#comment-50592Minor 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.]

]]>By: Jacobian conjecture « Simple or not simple?
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-43359
Sun, 24 Jan 2010 13:05:57 +0000http://terrytao.wordpress.com/?p=1869#comment-43359[...] Tao, Terence (2009-03-07). “Infinite fields, finite fields, and the Ax-Grothendieck theorem”. What’s New. [...]
]]>By: Terence Tao
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-42789
Fri, 11 Dec 2009 21:15:09 +0000http://terrytao.wordpress.com/?p=1869#comment-42789Ah yes, that would work too :-)
]]>By: L Spice
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/#comment-42769
Wed, 09 Dec 2009 20:23:34 +0000http://terrytao.wordpress.com/?p=1869#comment-42769I’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?
]]>