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 *ĥ*_{f}(*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.