Timeline for Criteria for irreducibility of polynomial
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 28, 2012 at 19:33 | comment | added | David E Speyer | My above comment is not quite right. The question Zieve's group was working on is when $F(x)-G(y)$ has a factor of genus $\leq 1$. I don't know whether they know when there is a factorization with both factors of high genus. | |
| Aug 24, 2012 at 11:52 | comment | added | David E Speyer | "Under the assumption that the F and G are indepecomposable (indecomposable meaning the polynomial is not the composition of two polynomials) there is a complete answer know (over the complex numbers)." Mike Zieve's REU (an incredible group of 6 undergrads) recently announced that they have a classification for all $(F,G)$, without the indecomposable hypothesis. If the indecomposability issue turns out to be crucial for you, you might want to e-mail Zieve. | |
| Aug 24, 2012 at 11:16 | comment | added | user9072 | @Rurik: your are welcome. I added some initial ideas on how one might proceed in practise. In particular, if one cannot find nontrivial F and G one is done. And, so this might already exclude many cases. And then for the (ir)reducibilty of F - G the question given in the comment by Camilo Sarmiento contains interesting information. Depending on the precise circumstances it could well be more direct to use such general criteria for two variable polys, then more specific one. But indeed this question, which I did not know before, contains also some info on precisely this type of polynomial. | |
| Aug 24, 2012 at 11:09 | history | edited | user9072 | CC BY-SA 3.0 |
typos corrected, some info added, some clean-up; added 20 characters in body
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| Aug 24, 2012 at 7:45 | comment | added | Rurik | Wow, that is really interesting. I do not know how easy it will be to apply this criteria (I have to check the irreducibility of 27 polynomial over the complex numbers and I am reclutant to use a computer algebra system since over the complex number they use approximations and therefore their answer cannot really considered as a PROOF of irreducibility...:D) but I will try anyway. Thank you very much for your very useful and detailed anwer, best | |
| Aug 24, 2012 at 7:06 | vote | accept | Rurik | ||
| Aug 23, 2012 at 16:12 | comment | added | user9072 | ...then by Siegel's theorem only finitely many solutions; thus the question when reducible; and then they study this; giving a nontrivial case of reducibility and conditions when irreducible. So, I think this how they came to this type of problems (but this is a guess). Several classical Dioph Equ fall into this category f(x)=g(y). | |
| Aug 23, 2012 at 16:00 | comment | added | user9072 | @Igor Rivin: The direct motivation given in the paper is that this question was raised by one of them (Schinzel) earlier: Some unsolved problems on polynomials, Matematicka Biblioteka 25 (1963), 63-70. Now, for the actual motivation, I think that this derives ultimately mainly from Diophantine Equations/Geometry. For example Davenport, Lewis, Schinzel wrote together somewhat earlier (1961) a paper "Equations of the form f(x)=g(y)" where the f,g are polynomials (with integral coefficients), and while I cannot see the paper the MR review says roughly that if irreducible and genus cond... | |
| Aug 23, 2012 at 15:23 | comment | added | Igor Rivin | That's quite surprising! What was the motivation for the Davenport/Schinzel work? | |
| Aug 23, 2012 at 14:38 | history | answered | user9072 | CC BY-SA 3.0 |