with Robert, Benedetto, Ruqian Chen, Trevor Hyde, and Colin White,
"Small Dynamical Heights for Quadratic Polynomials and Rational Functions"
(Experimental Mathematics, accepted.)
Abstract: 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 ĥf(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.