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

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