Linear relations with conjugates of a Salem number (Q2199678)

A non-zero algebraic number \(\alpha\) with conjugates \(\alpha_{1},\dots,\alpha _{d},\) is said to satisfy a non-trivial additive linear relation, in short NTALR (resp., a trivial additive linear relation, in short TALR), if there is \((k_{1},\dots,k_{d})\in \mathbb{Z}^{d}\) such that \[ k_{1}\alpha _{1}+\cdot \cdot \cdot +k_{d}\alpha _{d}=0, \] and \(k_{i}\neq k_{j}\) for some \(1\leq i<j\leq d\) (resp. and \(k_{1}=\cdots =k_{d}\neq 0).\) Clearly, algebraic numbers satisfying TALRs are those with trace \(0.\) Initiated by \textit{V. A. Kurbatov} [Mat. Sb., N. Ser. 43(85), 349--366 (1958; Zbl 0080.01404)], several authors have investigated the algebraic numbers satisfying NTALRs; see for instance [\textit{C. J. Smyth}, J. Number Theory 23, 243--254 (1986; Zbl 0586.12001)] where some related results have been proven and many conjectures have been formulated. In the present paper the authors consider the case where \(\alpha \) is a Salem number. They show, in this situation, that \(\alpha \) satisfies a NTALR (resp. a TALR) if and only if \((\alpha +1/\alpha )\) satisfies a NTALR (resp. a TALR). This allows them to obtain from a result of \textit{V. A. Kurbatov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1977, No. 1(176), 61--66 (1977; Zbl 0356.12027)] that the degree of a Salem number, satisfying a NTALR, is not twice a prime number. In addition, they give some examples of Salem numbers, with low degrees (including the smallest possible one, i.e., 8) satisfying NTALRs. Finally, using the same approach as in [\textit{C. J. Smyth}, Math. Comput. 69, No. 230, 827--838 (2000; Zbl 0988.11050)], they prove that for any even degree \(d\geq \) \(6\) there exists a Salem number of degree \(d\) satisfying a TALR.