Timeline for What is the geometry of an undecidable diophantine equation?
Current License: CC BY-SA 3.0
12 events
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| Jan 26, 2018 at 15:53 | comment | added | Timothy Chow | @XL_at_China : I don't know exactly what I mean by "most" which is why I used scare quotes. This is just an informal intuition, that there ought to be some definition of "most" according to which such a statement is true. | |
| Jan 26, 2018 at 1:22 | comment | added | XL _At_Here_There | What do you you mean by " "most" computably enumerable sets are not computable.", especially "most"? recursive sets are computablly enumerable, and so do c.e. sets that are not recursive. How to define your "most"? discrete measure? | |
| Dec 5, 2012 at 6:42 | comment | added | Will Sawin | By the way, I think in many reasonable measures a positive proportion of algebraic varieties have no solutions mod $p$, for any fixed $p$. So I don't think undecidable is quite the same as random. | |
| Dec 5, 2012 at 6:34 | comment | added | Will Sawin | That seems very reasonable to me. But perhaps it is very difficult but not impossible? I suppose there is only one way to find out. | |
| Dec 5, 2012 at 1:46 | comment | added | Timothy Chow | One small correction: If you're allowing families of systems then you don't have to encode a particular axiomatic system, but the members of the family still need to encode an undecidable problem, so I think it's still very difficult to ensure some nice geometric property at the same time. | |
| Dec 5, 2012 at 1:29 | comment | added | Timothy Chow | @Will: For what you're looking for, I think you need (1) explicit examples that are known to be undecidable, right? The trouble is that the only known way to construct such things is to encode the axioms that your system's lack of solutions can't be proven from. Though I don't know the state of the art, I suspect that it's very difficult to enforce some nice geometric property at the same time that you're encoding the axioms of ZFC (or whatever) in your equations. | |
| Dec 4, 2012 at 22:46 | comment | added | Will Sawin | sets of nice properties that do not have decidable Diophantine problems. I am interested in the second approach. I am disheartened by the dimension difficulty. That might entirely kill this question, but hopefully there is something interesting that can still be said. | |
| Dec 4, 2012 at 22:44 | comment | added | Will Sawin | I don't think this accurately reflects my question. There are many ways in which an algebraic variety can be special: low dimension, low kodaira dimension, etc. I would like to know about algebraic varieties which are "general" in that there diophantine problems are undecidable, but "special" in that they have some other nice property. There are two ways at getting at the existence of these things: showing that varieties with a certain set of nice properties have decidable Dipohantine problems (I'm aware of some conjectures to this effect), and the reverse: finding varieties with certain | |
| Dec 4, 2012 at 22:35 | comment | added | Timothy Chow | @David: Jones's system has 28 variables and 18 equations in its original formulation. Of course this doesn't necessarily mean that the dimension is 10. I don't know how to determine the dimension without first computing a Groebner basis. | |
| Dec 4, 2012 at 21:30 | comment | added | David E Speyer | Number of variables and degree is a clumsy way of describing geometry. I know that people in this field like to write systems $f_1=f_2=\cdots=f_N=0$ as a single system $f_1^2+f_2^2+\cdots + f_N^2$. If we undo all such obvious tricks to get back to a system of equations, then what is the dimension of the complex solution space? | |
| Dec 4, 2012 at 20:46 | comment | added | Steve Huntsman | This reminds me of the tone of Weinberger's book "Computers, Rigidity, and Moduli": books.google.com/books?id=CtvmQiuOKSEC | |
| Dec 4, 2012 at 20:17 | history | answered | Timothy Chow | CC BY-SA 3.0 |