The absolute trace of totally positive reciprocal algebraic integers (Q311457)

In the paper the authors prove that the trace of a totally positive reciprocal algebraic integer \(\alpha\) divided by its degree \(d\) exceeds \(1.8945909\dots\), except when the minimal polynomial of \(\alpha\) is \(x^2 - 3x + 1\), \(x^4 - 7x^3 + 13x^2 - 7x + 1\), or \(x^8 -15x^7 + 83x^6 - 220x^5 + 303x^4 - 220x^3 + 83x^2 - 15x + 1.\) In the proof they use the method of auxiliary functions. One should say that the conjectural best possible constant for totally positive algebraic integers is \(2-\varepsilon\), although J.-P.~Serre has shown that any the bound that is found by the method of explicit auxiliary functions cannot be beyond the value \(1.8983021\).