The lowest degree $0,1$-polynomial divisible by cyclotomic polynomial

TL;DR

This paper investigates the minimal degree of sparse polynomials with coefficients 0 or 1 that vanish at primitive roots of unity, focusing on cases with at most three prime factors.

Contribution

It characterizes the lowest-degree 0,1-polynomials divisible by cyclotomic polynomials for certain composite orders.

Findings

Identifies minimal degree polynomials for specific n

Provides bounds on polynomial degrees

Analyzes the structure of vanishing sums of roots of unity

Abstract

Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least three terms such that $f(\ze_n)$ is a minimal vanishing sum of the $n$-th roots of unity.