Small dynamical heights for quadratic polynomials and rational functions

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Let f ∈ Q(z) 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 f is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height hf(x) of a non-preperiodic rational point x is bounded below by a uniform multiple of the height of f 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.