Preperiodic points of polynomials over global fields

Given a global field K and a polynomial f defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of f is bounded in terms of only the degree of K and the degree of f. 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 f. By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of slog(s). Our bound applies to polynomials of any degree (at least two) over any global field K.