Timeline for Is the number of representations as the sum of two elements of a polynomial sequence always small?
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17 events
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| Mar 6, 2016 at 23:06 | answer | added | Brandon Hanson | timeline score: 2 | |
| Mar 3, 2016 at 15:30 | comment | added | Bobby Grizzard | Worth pointing out?: using the main theorem of Gaël Rémond's 2011 paper "Borne polynomiale pour le nombre de points rationnels des courbes" jtnb.cedram.org/item?id=JTNB_2011__23_1_251_0 gives that the number of rational points on the curve is at most $n^{2^{3^{D^2}}},$ where $D$ is the degree of $f$, again once the degree is large enough to make the genus $\geq 2$. | |
| Mar 3, 2016 at 15:03 | comment | added | Bobby Grizzard | @TerryTao correct. I should have said more. And I suppose the uniform boundedness conjecture about curves of fixed genus $g \geq 2$ is a famous enough conjecture that I should have said it would follow from that, but I don't know I nice name for that conjecture. | |
| Mar 3, 2016 at 5:06 | comment | added | Terry Tao | I assume Bobby Grizzard is referring to the result of Caporaso-Harris-Mazur that shows that Bombieri-Lang gives a uniform bound on the rational points of any curve of a fixed genus greater than or equal to 2: ams.org/mathscinet-getitem?mr=1325796 . This would indeed seem to give GH's claim, conditional of course on Bombieri-Lang. | |
| Mar 3, 2016 at 3:08 | comment | added | Mark Lewko | I'm curious if an improvement on the divisor bound is known for, say, $f(x)=x^5$? | |
| Mar 3, 2016 at 2:51 | comment | added | Bobby Grizzard | @GH, are you saying that because then the genus is at least 2, and boundedness would follow from the Bombieri-Lang Conjecture, or is there something more concrete that would apply to these specific curves? | |
| Mar 3, 2016 at 2:10 | comment | added | GH from MO | I would even think that $r(n)$ is bounded when $d$ is at least five. | |
| Mar 2, 2016 at 23:29 | comment | added | Mark Lewko | @Bobby, yes the constant can depend on $f$. | |
| Mar 2, 2016 at 23:20 | comment | added | Bobby Grizzard | The implied constant in the bound you seek can depend on $f$, correct? | |
| Mar 2, 2016 at 22:05 | comment | added | Richard Stanley | A related question is mathoverflow.net/questions/45511. | |
| Mar 2, 2016 at 19:19 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| Mar 2, 2016 at 6:48 | comment | added | Pig | Just out of curiosity, is there a story (similar to classical theta functions, along the lines of Fourier coefficient of modular forms) for series of the shape $\sum_{n \in \mathbb{Z}} \exp(2\pi i f(n)x)$ where $f(x) \in \mathbb{Z}[x]$ is a degree $d > 2$ polynomial? | |
| Mar 2, 2016 at 6:24 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| Mar 2, 2016 at 6:18 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| Mar 2, 2016 at 6:10 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| Mar 2, 2016 at 5:14 | history | edited | Mark Lewko |
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| Mar 2, 2016 at 5:06 | history | asked | Mark Lewko | CC BY-SA 3.0 |