Preperiodic points of polynomials over global fields

Abstract

Given a global field K and a polynomial defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational pre-periodic points of is bounded in terms of only the degree of K and the degree of . In 1997, for quadratic polynomials over K = ℚ, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of . By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of s log s. Our bound applies to polynomials of any degree (at least two) over any global field K.