Preperiodic Points of Polynomials over Global Fields

Given a global field K and a polynomial φ defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of φ is bounded in terms of only the degree of K and the degree of φ. 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 φ. 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. Let K be a field, and let φ ∈ K(z) be a rational function. Let φ denote the n iterate of φ under composition; that is, φ is the identity function, and for n ≥ 1, φ = φ ◦ φ. We will study the dynamics φ on the projective line P(K). In particular, we say a point x is preperiodic under φ if there are integers n > m ≥ 0 such that φ(x) = φ(x). The point y = φ(x) satisfies φ(y) = y and is said to be periodic, as its iterates will forever cycle through the same finite sequence of values. Note that x ∈ P(K) is preperiodic if and only if its orbit {φn(x) : n ≥ 0} is finite. For example, let K = Q and φ(z) = z2−29/16. Then {5/4,−1/4,−7/4} forms a periodic cycle (of period 3), and −5/4, 1/4, 7/4, and ±3/4 each land on this cycle after one or two iterations. In addition, the point ∞ is of course fixed. These nine Q-rational points are all preperiodic under φ. Meanwhile, it is not difficult to see that no other point in P(Q) is preperiodic by showing that the denominator of a rational preperiodic point must be 4, and that the absolute value must be less than 2. In general, for any global field K, any dimension N ≥ 1, and any morphism φ : P → P over K of degree at least two, Northcott proved in 1950 that the number of K-rational preperiodic points of φ is finite [25]. More precisely, he showed that the preperiodic points form a set of bounded arithmetic height. Years later, by analogy with the Theorems of Mazur [19] and Merel [20] on K-rational torsion of elliptic curves, Morton and Silverman proposed the following Conjecture [23]. Uniform Boundedness Conjecture. (Morton and Silverman, 1994) Given integers D,N ≥ 1 and d ≥ 2, there is a constant κ = κ(D,N, d) with the following property. Let K be a number field with [K :Q] = D, and let φ : P → P be a morphism of degree d defined over K. Then φ has at most κ preperiodic points in P(K). The analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve E are precisely the preperiodic points of the multiplication-bytwo map [2] : E → E. In fact, taking x-coordinates, the map [2] induces a rational function Date: June 21, 2005. 2000 Mathematics Subject Classification. Primary: 11G99 Secondary: 11D45, 37F10.