On Salem numbers, expansive polynomials and Stieltjes continued fractions

TL;DR

This paper explores a new method linking Salem numbers, expansive polynomials, and Stieltjes continued fractions, revealing algebraic structures and providing a converse to Salem's original construction.

Contribution

It introduces a novel association between Salem polynomials and Hurwitz quotients via expansive polynomials, expanding the understanding of Salem number construction.

Findings

Salem numbers can be characterized by Stieltjes continued fractions.

A semigroup structure exists among certain Salem polynomials and Garsia numbers.

The method provides a converse to Salem's original construction of Salem numbers.

Abstract

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $\beta$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over a Salem polynomial having simple roots, not cancelling at $\pm 1$, coming from monic expansive polynomials of constant term equal to their Mahler measure, has a semigroup structure. The sets of corresponding generalized Garsia numbers inherit this semi-group structure.