The measure of totally positive algebraic integers

Abstract

For a totally positive algebraic integer α ≠ 0 , 1 of degree d , we consider the set R of values of L ( α ) 1 / d = R ( α ) and the set L of values of M ( α ) 1 / d = Ω ( α ) , where L ( α ) is the length of α and M ( α ) is the Mahler measure of α . In this paper, we prove that all except finitely many totally positive algebraic integers α have R ( α ) ⩾ 2.364950 and Ω ( α ) ⩾ 1.721916 . The computation uses a family of explicit auxiliary functions. We notice that several polynomials with complex roots are used to construct the functions. We also find eight totally positive irreducible polynomials with absolute length greater than 2.364950 and less than 2.37.