A characterization of two related classes of Salem numbers
If \(P\) is a monic polynomial with integer coefficients and \(P^* (z)= z^{\deg P} P(1/z)\) is its reciprocal then sometimes it happens that the polynomial \(Q(z)= zP(z)+ \varepsilon P^* (z)\) (where \(\varepsilon= \pm1\)) has a single zero outside the unit circle. If that zero is positive and not a root of \(z^ 2- az+1\) for some \(a>2\) then it is a Salem number. If one adds the restriction \(| P(0) |=q\) (for fixed \(q\)) and \(\varepsilon=- \text{sgn } P(0)\) then the set of so obtained Salem numbers is denoted by \(A_ q\). The change of the second restriction to \(\varepsilon= \text{sgn } P_ 0 (0)\), where \(P(z)= z^ m P_ 0 (z)\), \(m>0\) and \(P_ 0 (0)\neq 0\) leads to a set \(B_ q\) of Salem numbers. These sets have been considered earlier by the first author [Acta Arith. 39, 207-240 (1981; Zbl 0514.12020)] resp. the second author [Math. Comput. 35, 1361-1377 (1980; Zbl 0447.12002)]. The authors prove a characterization of elements of \(A_ q\) and \(B_ q\) among Salem numbers which imply \(A_ q \subset B_{q-2}\) for \(q\geq 2\).
scientific article; zbMATH DE number 729081