Timeline for Integral points on genus 0 curves related to polynomial identities
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Apr 3, 2023 at 18:38 | comment | added | Peter Mueller | ... Note that indeed this function need not be a polynomial. Take for instance $1/(x^2-2)$. Then its values are integers at $u/v$ where $u,v$ satisfy the Pell equation $u^2-2v^2=1$. These things are handled in Lang's Diophantine Geometry, Chapt. VII, Section 4 (Curves of genus $0$), and probably in his newer edition Fundamentals of diophantine geometry too. | |
| Apr 3, 2023 at 18:34 | comment | added | Peter Mueller | @AlexM. Unfortunately, both sources you link to miss the genus $0$ case, which itself is interesting and highly nontrivial. If there are sufficiently many rational points on the curve, then this curve has a rational parameterization over the rationals. So the problem essentially reduces to the following: What can we say about a rational function which assumes integer values for infinitely many rational arguments? Siegel proved the necessary condition that this function has at most two poles ... | |
| Apr 3, 2023 at 14:47 | comment | added | Alex M. | Is this Siegel's theorem that you are using (another source is here)? I am asking because its statement is given only for genus $>0$, whereas the OP claims that his curve has genus $0$. | |
| Aug 5, 2013 at 12:27 | history | answered | Peter Mueller | CC BY-SA 3.0 |