Comments on Salem polynomials
The author gives a sufficient condition on a monic irreducible reciprocal polynomial \(P \in \mathbb{Z}[x]\) of even degree \(d \geq 4\) to have \(d-2\) roots on the unit circle, refining a result of \textit{R. S. Vieira} [Ramanujan J. 42, No. 2, 363--369 (2017; Zbl 1422.30013)]. More precisely, write \(P = x^d + c_1 x^{d-1} + \cdots + c_1 x + 1\), and denote by \(L\) be the length of \(P\), that is, the sum of the absolute values of the coefficients of \(P\). The author shows that if \(|c_1| > L/4\), then \(P\) has \(d-2\) roots on the unit circle, and two real roots \(\alpha\) and \(\alpha^{-1}\), with \(|\alpha|>1\). In particular, \(|\alpha|\) is a Salem number. Furthermore, the roots of \(P\) on the unit circle interlace with the \((d-2)\)-th roots of unity. The proof is a simple application of the intermediate value theorem and Hurwitz's theorem in complex analysis. The author also gives a variant of a result of \textit{D. Stankov} [J. Théor. Nombres Bordx. 31, No. 1, 215--226 (2019; Zbl 1453.11135)], by characterising Salem polynomials among reciprocal polynomials, with no irreducibility assumption.
scientific article; zbMATH DE number 7371067
scientific article