The Uniform Boundedness and Dynamical Lang Conjectures for polynomials
Nicole R. Looper
TL;DR
The paper develops a global, quantitative equidistribution framework for polynomial dynamics over number fields and function fields, bridging non-archimedean potential theory with higher-dimensional height conjectures. By introducing wing decompositions and ε-equidistribution, it shows that large sets of points with small canonical heights must exhibit controlled distribution across bad reduction places, enabling a contradiction with the abcd conjecture when too many such points exist. Under the abcd conjecture, it establishes a uniform bound on the number of K-rational preperiodic points for degree d polynomials and a dynamical Lang-type lower bound for canonical heights, with unconditional results available in certain function-field cases (notably unicritical polynomials of degree ≥ 5). The work thus extends prior unicritical results to general polynomials and provides a robust toolkit for approaching dynamical generalizations of Lang and related height conjectures.
Abstract
We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. 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.