Timeline for What is the geometry of an undecidable diophantine equation?

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Dec 5, 2012 at 16:44	comment	added	Timothy Chow		@Misha: The content of the MRDP theorem is that every computably enumerable set is Diophantine. This is a lot stronger than the "mere" statement that it's undecidable whether a Diophantine system has a solution. Now, it's true that converting a computably enumerable set to a Diophantine set is a somewhat delicate process, so it's not ruled out that there still some sense in which "generic" Diophantine sets are decidable. I think it would be pretty surprising, though.
Dec 5, 2012 at 6:38	comment	added	Will Sawin		I think the Hasse principle is very strongly related to decidability, but I'm not sure if it always implies decidability.
Dec 5, 2012 at 6:35	comment	added	Will Sawin		I don't know much about explicit decidability results. Here is an interesting discussion of a conjectural one: `http://terrytao.wordpress.com/2007/05/04/distinguished-lecture-series-iii-shou-wu-zhang-%E2%80%9Ctriple-l-series-and-effective-mordell-conjecture%E2%80%9D/`
Dec 5, 2012 at 4:55	comment	added	Misha		@Will: Could you add some (interesting) examples of geometric conditions implying solvability of Diophantine equations to your question? I think, it would be quite valuable (for people like me). One thing I find surprising is Timothy Chow's comment that unsolvability is expected to be generic: This contradicts my intuition coming from group theory.
Dec 5, 2012 at 4:15	comment	added	Will Sawin		I think you're right that the other way makes more sense, but I think there is a lot of work of this nature, even if it doesn't make explicit references to computation. Methods for solving Diophantine equations are a huge, huge field, and I already know many key ideas. That's why I'm interested in the reverse.
Dec 5, 2012 at 4:00	history	answered	Misha	CC BY-SA 3.0