Salem numbers of trace −2, and a conjecture of Estes and Guralnick

In 1993 Estes and Guralnick conjectured that any totally real separable monic polynomial with rational integer coefficients will occur as the minimal polynomial of some symmetric matrix with rational integer entries. They proved this to be true for all such polynomials that have degree at most 4.In this paper, we show that for every  d≥6d≥6 there is a polynomial of degree d that is a counterexample to this conjecture. The only case still in doubt is degree 5.One of the ingredients in the proof is to show that there are Salem numbers of degree 2d   and trace −2 for every d≥12d≥12.