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The Remak height for units

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Abstract

We investigate the values of the Remak height, which is a weighted product of the conjugates of an algebraic number. We prove that the ratio of logarithms of the Remak height and of the Mahler measure for units αof degree d is everywhere dense in the maximal interval [d/2(d-1),1] allowed for this ratio. To do this, a “large” set of totally positive Pisot units is constructed. We also give a lower bound on the Remak height for non-cyclotomic algebraic numbers in terms of their degrees. In passing, we prove some results about some algebraic numbers which are a product of two conjugates of a reciprocal algebraic number.

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  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania

    A. Dubickas

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  1. A. Dubickas
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Dubickas, A. The Remak height for units. Acta Mathematica Hungarica 97, 1–13 (2002). https://doi.org/10.1023/A:1020822326977

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