THERE ARE SALEM NUMBERS OF EVERY TRACE

Abstract

The existence of Salem numbers of every trace is proved; the nontrivial part of this result concerns Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, has enabled the construction to be refined so as to avoid cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. It is also shown how this new interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbitrary trace than was previously known.