The ${\rm R}_2$ measure for totally positive algebraic integers

Abstract

Let α be a totally positive algebraic integer of degree d, i.e.,whose all conjugates α 1 = α,. . ., α d are positive real numbers. We study the set R 2 of the quantities (d i=1 (1 + α 2 i) 1/2) 1/d. We first show that √ 2 is the smallest point of R 2. Then, we prove that there exists a number l such that R 2 is dense in (l, ∞). Finally, using the method of auxiliary functions, we find the six smallest points of R 2 in (√ 2, l). The polynomials involved in the auxiliary function are found by our recursive algorithm.