Binary polynomial power sums vanishing at roots of unity
Volume 198 / 2021
Acta Arithmetica 198 (2021), 195-217
MSC: Primary 11D61; Secondary 11G50, 11J86.
DOI: 10.4064/aa200511-12-9
Published online: 21 December 2020
Abstract
Let ${c_1(x),c_2(x),f_1(x),f_2(x)}$ be polynomials with rational coefficients. The “obvious” exceptions being excluded, there can be at most finitely many roots of unity among the zeros of the polynomials ${c_1(x)f_1(x)^n+c_2(x)f_2(x)^n}$ with $n=1,2\ldots .$ We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.
