# 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.

## Key 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.