The trace of totally positive algebraic integers

Abstract

For all totally positive algebraic numbers α \alpha except a finite number of explicit exceptions, the following inequality holds: \[ 1 d ( α 1 + ⋯ + α d ) > max ( 1.780022 , 1.66 + α 1 ) , \frac {1}{d}\,(\alpha _1+\dots +\alpha _d)>\max (1.780022,1.66+\alpha _1), \] where d d is the degree of α \alpha and 0 > α 1 > ⋯ > α d 0>\alpha _1>\dots >\alpha _d its conjugates. This improves previous results of Smyth, Flammang and Rhin.