Some conjectures on the maximal height of divisors of $x^n-1$

TL;DR

This paper investigates the maximum height of divisors of cyclotomic polynomials, proposing conjectures for specific cases and establishing a lower bound for certain prime power products.

Contribution

It introduces conjectures on B(n) for specific n and proves a lower bound for B(p^a q^b) where p and q are distinct primes.

Findings

Formulated conjectures on B(n) for particular n

Proved a lower bound for B(p^a q^b) with prime p,q

Enhanced understanding of divisor heights of cyclotomic polynomials

Abstract

Define B(n) to be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. We formulate a number of conjectures related to the value of B(n) when $n$ is of a prescribed form. Additionally, we prove a lower bound for $B(p^aq^b)$ where $p,q$ are distinct primes.