Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Abstract

Let be a polynomial or rational function of degree 2. A special case of Morton and Silverman’s dynamical uniform boundedness conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height of a nonpreperiodic rational point x is bounded below by a uniform multiple of the height of φ itself. We provide support for these conjectures by computing the set of preperiodic and small-height rational points for a set of degree-2 maps far beyond the range of previous searches.