The Remak height for units. (English) Zbl 1059.11062
Summary: 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 \(\alpha\) of 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. Further, we prove some results about some algebraic numbers which are a product of two conjugates of a reciprocal algebraic number.
MSC:
11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
11R09 | Polynomials (irreducibility, etc.) |
11R80 | Totally real fields |