On the coefficients of divisors of $x^n-1$

TL;DR

This paper extends Suzuki's theorem by showing that for divisors of $x^n-1$ with a fixed number of irreducible factors, any finite integer sequence can be realized as coefficients, and provides bounds for the maximum coefficient magnitude.

Contribution

It generalizes Suzuki's result to divisors with a specified number of irreducible factors and establishes tight bounds for their coefficients.

Findings

Any finite integer sequence can be realized as coefficients of some divisor of $x^n-1$ with fixed irreducible factors.

Established tight bounds for the maximum absolute value of coefficients of such divisors.

Extended the understanding of the coefficient structure of divisors of cyclotomic-related polynomials.

Abstract

Let $a(r,n)$ be $r$th coefficient of $n$th cyclotomic polynomial. Suzuki proved that $\{a(r,n)|r\geq 1,n\geq 1\}=\mathbb{Z}$. If $m$ and $n$ are two natural numbers we prove an analogue of Suzuki's theorem for divisors of $x^n-1$ with exactly $m$ irreducible factors. We prove that for every finite sequence of integers $n_1,\ldots,n_r$ there exists a divisor $f(x)=\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\in \mathbb{N}$ such that $c_i=n_i$ for $1\leq i \leq r$. Let $H(r,n)$ denote the maximum absolute value of $r$th coefficient of divisors of $x^n-1$. In the last section of the paper we give tight bounds for $H(r,n)$.