Abstract

In this note, we study the root structure of the family of polynomials p(k,r)(x) = x(k) - x(k-1) - & centerdot;& centerdot;& centerdot; - x (R) + x(r-1) + & centerdot;& centerdot;& centerdot; + x + 1, with k > r >= 0 and k >= 4. We prove that for r = 1, its dominant root is a Salem number, and the cyclotomic polynomial Phi(n)(x) is a factor if and only if n = 6 and k equivalent to 2 (mod 6). For k odd we prove that the polynomial p(k,1)(x)/(x + 1) is irreducible. Moreover, if r >= 2 and k >= 2r + 2 then the dominant root of p(k,r)(x) is neither Pisot nor Salem. Solving a conjecture stated by San Mart & iacute;n and Sirvent in [B. San Mart & iacute;n, V.F. Sirvent, Chaos Solitons Fractals 169 (2023) 11. Paper No. 113239]. The case r + 1 <= k <= 2r, is also studied. Furthermore, we characterized all the cyclotomic factors of pk,r, for r >= 1.