Comments on: Ax’s Theorem: An Application of Logic to Ordinary Mathematics http://xorshammer.com/2008/08/15/axs-theorem/ Some things in mathematical logic that I find interesting Sat, 03 Mar 2012 11:53:40 +0000 hourly 1 http://wordpress.com/ By: Joshua Zelinsky http://xorshammer.com/2008/08/15/axs-theorem/#comment-423 Tue, 08 Mar 2011 15:56:47 +0000 http://xorshammer.wordpress.com/?p=28#comment-423 This is a very nice theorem. One thing that I’ve been wondering about in this context is whether there are any examples of injective polynomials f that aren’t surjective.

Incidentally, one way to think of this theorem is as a generalization of the fundamental theorem of algebra since in the single variable case all polynomials are surjective. Similarly, one gets from Picard’s theorem that all injective analytic functions are surjective (consider f composed with itself).

]]> By: Cups, Peas, and Grothendieck « Gödel’s Lost Letter and P=NP http://xorshammer.com/2008/08/15/axs-theorem/#comment-236 Tue, 16 Feb 2010 14:17:26 +0000 http://xorshammer.wordpress.com/?p=28#comment-236 [...] called the Ax-Grothendieck theorem—after James Ax and Grothendieck. I love this beautiful theorem and am currently trying to extend it. My combinatorial problem is a lemma that I need to prove my [...]

]]> By: Mathematical Embarrassments « Gödel’s Lost Letter and P=NP http://xorshammer.com/2008/08/15/axs-theorem/#comment-229 Sat, 26 Dec 2009 14:14:07 +0000 http://xorshammer.wordpress.com/?p=28#comment-229 [...] Amazing. The full proof is based on a simple compactness argument and this simple observation. See this for a nice exposition by Michael [...]

]]> By: rjlipton http://xorshammer.com/2008/08/15/axs-theorem/#comment-224 Wed, 16 Dec 2009 03:02:49 +0000 http://xorshammer.wordpress.com/?p=28#comment-224 I linked to your proof. One of my favorite theorems/proofs. So cool that such a simple idea can be made to work.

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