ABC implies primitive prime divisors in arithmetic dynamics

By Chad Gratton, K. Nguyen, T. Tucker. 2012

63 citations recorded by the source corpus.

Let K be a number field, let φ(x)∈K(x) be a rational function of degree d>1, and let α∈K be a wandering point such that φn(α)≠0 for all n>0. We prove that if the abc‐conjecture holds for K, then for all but finitely many positive integers n, there is a prime 픭 of K such that v픭(φn(α))>0 and v픭(φm(α))⩽0 for all positive integers m0 with the stronger condition v픭(φn(α))=1. We prove the same result unconditionally for function fields of characteristic 0 when φ is not isotrivial.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6e43a289eb57c6900a70890785906f9f78f4d90

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In Geometry and Topology

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Citations in the corpus

Current trends and open problems in arithmetic dynamics (2018, background): “There are stronger conditional results if one assumes the ABC-conjecture [102,158] or Vojta’s conjecture [219].”