On a number theoretical application of Coxeter transformations (Q2726133)

Let \(d, m \geq 2\) be two positive integers. The author shows that the polynomial \(z^d-(m-1)(z^{d-1}+ \dots +z+1)\) defines a Pisot number \(\nu\) and proves that \(m-m^{1-d} < \nu < m- m^{-d}.\) The proof is elementary. The above polynomial is related to the spectral radii of the Coxeter transformation of a certain connected graph. NEWLINENEWLINENEWLINEThe same polynomial, although for different reasons, was considered earlier by many authors, including \textit{W. J. Floyd} [Math. Ann. 293, No.~3, 475-483 (1992; Zbl 0735.51016)], the reviewer [Liet. Mat. Rink. 39, No.~3, 310-316 (1999; Zbl 0972.11013)] and others. For instance, Theorem 3 in the reviewer's paper [Can. Math. Bull. 45, No.~2, 196-203 (2002)] is essentially the same result as that of this paper, but the proof, although being much shorter, is not elementary, since it involves Rouché's theorem. NEWLINENEWLINENEWLINEThere is a misprint on p. 295: the smallest known Salem number is \(1.1762 \dots,\) and not \(1.3241 \dots.\)