For an algebraic number α we denote by M (α) the Mahler measure of α. As M (α) is again an algebraic number (indeed, an algebraic integer), M (·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #O M (α), is the cardinality of the forward orbit of α under M . We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size.
In this note we investigate the behaviour of the absolute logarithmic Weil-height h on extensions of the field Q tr of totally real numbers. It is known that there is a gap between 0 and the next smallest value of h on Q tr , whereas in Q tr (i) there are elements of arbitrarily small positive height. We prove that all elements of small height in any finite extension of Q tr already lie in Q tr (i). This leads to a positive answer to the question of Amoroso, David and Zannier, if there exists a pseudo algebraically closed field with the mentioned height gap.
The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than log(3) /p.
In 1973 Schinzel proved in [20] 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.
Abstract. We study the behavior of canonical height functions h f , associated to rational maps f , on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of h f on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f (X) = X for some non-linear polynomial f . This answers a question of W.
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