Merel’s uniform boundedness theorem for torsion points on elliptic curves inspired Morton and Silverman's dynamical uniform boundedness conjecture, whose general form would imply the uniform boundedness of torsion points on abelian varieties. In this talk, I will discuss an effective uniform bound for periodic points of the polynomials z^d + c over every number field K when d ≥ 4, and over every number field K not containing ℚ(√−3) when d = 3. The proof combines Kummer theory with the Diophantine geometry of twisted Fermat curves. I will also explain how the remaining cases are related to the boundedness of Mordell–Weil ranks of certain families of elliptic curves and Jacobians.