Timeline for Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Current License: CC BY-SA 3.0
6 events
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| Feb 4, 2014 at 11:39 | comment | added | Lior Bary-Soroker | I forgot to say that a field $K$ is called ample if for every smooth geometrically connected curve $C$ over $K$, the set of rational points $C(K)$ is either empty or infinite. This property is clearly satisfied for your fields, and also for other fields like complete fields, etc. | |
| Feb 4, 2014 at 11:38 | comment | added | Lior Bary-Soroker | Koenigsmann proves this result in a more general setting. Namely he proves that if $K$ is ample, then the image of an irreducible separable polynomial of degree at most $2$ is NOT cofinite. See the argument in arXiv:1106.1310 after Conjecture 6.1. | |
| Jan 21, 2014 at 14:57 | comment | added | Michiel Kosters | Actually, with some changes I can prove the statement for a field $k$ with 1. $k$ is perfect 2. every geometrically irreducible normal projective curve over k has infinitely many k-points. The proof uses the statement that the exact sequence with the decomposition group and the inertia group splits. | |
| Nov 14, 2013 at 12:00 | history | edited | Michiel Kosters | CC BY-SA 3.0 |
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| Nov 6, 2013 at 21:16 | history | edited | Michiel Kosters | CC BY-SA 3.0 |
added 15 characters in body
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| Nov 6, 2013 at 13:25 | history | answered | Michiel Kosters | CC BY-SA 3.0 |