Dynamical uniform boundedness for unicritical polynomials and elliptic curves of large rank

Abstract

For every number field $K$, we prove the dynamical uniform boundedness conjecture for the unicritical family of polynomials $z^d + c$ when $d \geq 4$, and when $d = 3$ if $\mathbb{Q}(\sqrt{-3}) \not\subset K$. Furthermore, the unconditional bounds that we construct for periodic points of unicritical polynomials are effectively computable. In all of the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over $K$ for a specific family of elliptic curves when $d = 3$ and for Jacobians of dynatomic curves when $d = 2$.