Binary polynomial power sums vanishing at roots of unity

TL;DR

This paper investigates the roots of unity among zeros of certain polynomial sums involving powers, providing bounds based on polynomial degrees and heights, and extends understanding of polynomial behavior at roots of unity.

Contribution

It offers new bounds on the orders of roots of unity for polynomial power sums, linking these bounds to polynomial degrees and heights.

Findings

Finitely many roots of unity occur in these polynomial sums.

Bounds on the orders of roots of unity are established.

Results depend on degrees and heights of the involved polynomials.

Abstract

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, 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 $f_i$.