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The Remak height for units
Authors:A. Dubickas
Affiliation:(1) Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania
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. This revised version was published online in June 2006 with corrections to the Cover Date.
Keywords:Mahler measure  Remak height  algebraic unit  Pisot numbers
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