Ana-Maria Stan, Florin Stan: Some remarks on totally positive algebraic integers, 373-379

Abstract:

In this paper we prove several results concerning the set $\mathbb{A}_{+}$ of totally positive algebraic integers. We prove that the set $\mathbb{A}_{+}$ is dense in the set of positive real numbers. We explicitly construct an infinite family of cubic polynomials, which are minimal polynomials of totally positive algebraic integers, and use it to show that the distance between a totally positive algebraic integer and one of its conjugates can be arbitrarily small. Finally, we employ a new method to construct, for any prime $p \ge 3$, a monic, integer, irreducible polynomial of degree $p-1$, with all roots positive.

Key Words: Totally positive algebraic integer, Schur-Siegel-Smyth trace problem.

2020 Mathematics Subject Classification: Primary 11R04; Secondary 11R06.

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