Heights of divisors of x^n-1

TL;DR

This paper investigates the maximal height of polynomial divisors of x^n-1, providing bounds analogous to Maier's results on cyclotomic polynomial heights, advancing understanding of polynomial divisor heights.

Contribution

It extends Maier's bounds on cyclotomic polynomial heights to the maximal height of all polynomial divisors of x^n-1, offering new asymptotic bounds.

Findings

Established bounds on the maximal height of polynomial divisors of x^n-1

Extended Maier's results to a broader class of polynomials

Provided asymptotic analysis of divisor heights

Abstract

The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best possible upper bound of n^{\psi(n)} for almost all n, where \psi(n) is any function that approaches infinity as n tends to infinity. We will discuss the related problem of bounding the maximal height over all polynomial divisors of x^n - 1 and give an analogue of Maier's result in this scenario.