TL;DR
This paper investigates the lower bounds of canonical heights in dynamical systems over totally real fields, extending Schinzel's classical result on the standard logarithmic height to a dynamical setting.
Contribution
It extends Schinzel's height lower bound results to canonical heights associated with rational functions in dynamical systems on the affine line.
Findings
Lower bounds for canonical heights in dynamical systems over totally real fields.
Behavior of height functions on finite extensions of maximal totally real fields.
Remarks on the properties of heights in the context of rational functions.
Abstract
1973 Schinzel proved that the standard logarithmic height h on the maximal totally real field extension of the rationals is either zero or bounded from below by a positive constant. In this paper we study this property for canonical heights associated to rational functions and the corresponding dynamical system on the affine line. At the end, we will give a few remarks on the behavior of h on finite extensions of the maximal totally real field.
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