Salem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers

Until recently, no Salem numbers were known of trace below −1. In this paper we provide several examples of trace −2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace −2 is 20, and exhibit all Salem numbers of degree 20 and trace −2. Indeed there are just two examples. We also settle the closely-related question of the minimal degree d of a totally positive algebraic integer such that its trace is ≤ 2d −2. This minimal degree is 10, and there are exactly three conjugate sets of degree 10 and trace 18. Their minimal polynomials enable us to prove that all except five conjugate sets of totally positive algebraic integers have absolute trace greater than 16/9. We end with a speculative section where we prove that, if a single polynomial with certain properties exists, then the trace problem for totally positive algebraic integers can be solved.