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Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.

Question: Is there any criterion(proved or conjectural) for the existence of integral points on such a curve?

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  • $\begingroup$ In the range of my knowledge, I suppose this question is very difficult to answer. Intuitively, the integral points must exist on the curve. $\endgroup$ – Jame Ake Oct 8 '13 at 11:12
  • $\begingroup$ The Integral points constitute an Two-dimensional lattice on the complex plane, so the question can be transformed into this: Are all the rational function cann't intersect with the Intergral Points? $\endgroup$ – Jame Ake Oct 8 '13 at 11:22
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    $\begingroup$ Conjecturally, if there is no integral Brauer-Manin obstruction then the curve has integral points. $\endgroup$ – Felipe Voloch Oct 8 '13 at 13:07
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    $\begingroup$ @Felipe Voloch: Is there a low-brow account of the integral Brauer-Manin obstruction which works practically in terms of the rational functions $f_1$ and $f_2$? $\endgroup$ – Peter Mueller Oct 10 '13 at 16:02
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    $\begingroup$ @PeterMueller Not that I know of. If you convert the problem into a unit equation in the standard way, then the Brauer-Manin obstruction can be turned into congruence conditions. $\endgroup$ – Felipe Voloch Oct 10 '13 at 16:11

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