ABC implies primitive prime divisors in arithmetic dynamics

作者：Gratton C^*; Nguyen K; Tucker T J

来源：Bulletin of the London Mathematical Society, 2013, 45(6): 1194-1208.

DOI：10.1112/blms/bdt049

Let K be a number field, let phi(x)is an element of K(x) be a rational function of degree d > 1, and let alpha is an element of K be a wandering point such that phi(n)(alpha)not equal 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 p of K such that v(p)(phi(n)(alpha))> 0 and v(p)(phi(m)(alpha))< 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition v(p)(phi(n)(alpha))> 0 with the stronger condition v(p)(phi(n)(alpha))=1. We prove the same result unconditionally for function fields of characteristic 0 when phi is not isotrivial.

• 出版日期2013-12