  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.
