There are Salem Numbers of Every Trace
Department of Mathematics, Royal Holloway, University of London Egham Hill, Egham Surrey TW20 0EX, United Kingdom; James.McKee{at}rhul.ac.uk
School of Mathematics, University of Edinburgh James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom; C.Smyth{at}ed.ac.uk
Received 21 August 2003. Revision received 24 February 2004.
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. 2000 Mathematics Subject Classification 11R06.