Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772648 | Journal of Number Theory | 2017 | 22 Pages |
Abstract
Let K be a number field and let Ï in K(z) be a rational function of degree dâ¥2. Let S be the set of places of bad reduction for Ï (including the archimedean places). Let Per(Ï,K), PrePer(Ï,K), and Tail(Ï,K) be the set of K-rational periodic, preperiodic, and purely preperiodic points of Ï, respectively. The present paper presents two main results. The first result is a bound for |PrePer(Ï,K)| in terms of the number of places of bad reduction |S| and the degree d of the rational function Ï. This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that |Per(Ï,K)|â¥4 (resp. |Tail(Ï,K)|â¥3), we prove bounds for |Tail(Ï,K)| (resp. |Per(Ï,K)|) that depend only on the number of places of bad reduction |S| (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when |Per(Ï,K)|<4 (resp. |Tail(Ï,K)|<3).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Sebastian Troncoso,