A Remark on the Effective Mordell Conjecture and Rational Pre-Images under Quadratic Dynamical Systems

TL;DR

This paper explores how the number of rational points in backward orbits of quadratic dynamical systems depends on the basepoint, assuming a strong form of the Mordell Conjecture, and discusses implications for the effective Mordell conjecture.

Contribution

It investigates the dependence of rational point bounds on the basepoint in quadratic dynamical systems under a strong Mordell Conjecture assumption.

Findings

Bound on rational points is independent of c for fixed b

Dependence of bounds on b is analyzed under conjectural assumptions

Implications for the effective Mordell conjecture are discussed

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.