2016/08/03 by Ghioca, Dragos, Nguyen, Khoa D., Tucker, Thomas J.
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1608.01361
Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pairs of integers (m,n) with m nonnegative and n positive, and show that for all but finitely many (m,n) in A(f,a) there is a prime p of K which is not in S such that the p-adic valuation of fm+n(a)-fm(a) is precisely equal to 1, and moreover a has portrait (m,n) under the action of f modulo p. This latter condition implies that the p-adic valuation of fu+v(a)-fu(a) is not positive if u is a nonnegative integer and v is a positive integer with u