UNIFORM BOUNDS ON PRE-IMAGES UNDER QUADRATIC DYNAMICAL SYSTEMS

摘要

For any elements a, c of a number field K, let Г(a, c) denote the backwards orbit of a under the map f_c: C( )C given by f_c(x) = x~2 + c. We prove an upper bound on the number of elements of Г(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also show that, for any fixed N ( ) 4 and any a _ K outside a finite set, there are only finitely many pairs (yo, c) _ C~2 for which [K(yo, c) : K] < 2~(N-3) and the value of the N~(th) iterate of f_c(x) at x = yo is a. Moreover, the bound 2~(N-3) in this result is optimal.