On the Remak Height, the Mahler Measure, and Conjugate Sets of Algebraic Numbers Lying on Two Circles

We deene a new height function R (), the Remak height of an algebraic number. We give sharp upper and lower bounds for R () in terms of the classical Mahler measure M () : Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm 1 lying on two circles centered at 0. We give a complete characterisation of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalisation of Salem numbers which we call extended Salem numbers. 1. Introduction. Let be an algebraic number, of degree d > 2, with minimal polynomial a 0 z d + ::: + a d 2 Z z] over the rationals, conjugates 1 ; 2 ; :::; d (with one of these) labelled so that j 1 j>j 2 j> ::: >j d j. In 1952 Robert Remak Re] gave a new upper bound for the