A remark on the effective Mordell Conjecture and rational pre-images under quadratic dynamical systems

Abstract

Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f c ( x ) = x 2 + c , it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short Note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.