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A remark on the effective Mordell conjecture and rational pre-images under quadratic dynamical systems. (English. French summary) Zbl 1264.37053
Summary: Fix a rational base point $$b$$ 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 base point $$b$$, assuming a strong form of the Mordell conjecture.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems 11G50 Heights 14G05 Rational points
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