Maximal height of divisors of $x^{pq^{b}}-1$

TL;DR

This paper investigates the maximal height of divisors of the polynomial $x^{pq^b}-1$, confirming some conjectures and advancing understanding of polynomial divisor heights in algebraic number theory.

Contribution

It proves the validity of certain conjectures regarding the maximal height of divisors of $x^{pq^b}-1$, expanding knowledge in polynomial divisor properties.

Findings

Confirmed conjectures on maximal divisor height

Established bounds for heights of divisors of $x^{pq^b}-1$

Enhanced understanding of polynomial divisor structures

Abstract

The height of a polynomial $f(x)$ is the largest coefficient of $f(x)$ in absolute value. Let B(n) be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. In this paper we investigate the maximal height of divisors of $x^{pq^b}-1$ and prove that some conjectures on the maximal height of divisors of $x^{pq^b}-1$ are true.