The Uniform Boundedness and Dynamical Lang Conjectures for polynomials

We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials. Our technique simultaneously yields a dynamical analogue of Lang's conjecture on minimal canonical heights for these maps. We obtain similar results for non-isotrivial polynomials over a function field of characteristic zero. When the latter are unicritical of degree at least 5, the results hold unconditionally.