On the height of cyclotomic polynomials

TL;DR

This paper establishes new bounds on the height of cyclotomic polynomials and their inverses, as well as on divisors of $x^n-1$, revealing exponential growth patterns in these bounds.

Contribution

It provides improved upper bounds on the heights of cyclotomic polynomials, their inverses, and divisors of $x^n-1$, with explicit exponential growth rates.

Findings

Bound on $A_n$ involving $ abla_k \

Bound on $C_n$ for inverse cyclotomic polynomials

Enhanced bound on $B_n$ for divisors of $x^n-1$

Abstract

Let $A_n$ denote the height of cyclotomic polynomial $\Phi_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le \epsilon_k\phi(n)^{k^{-1}2^{k-1}-1}$ with $-\log\epsilon_k\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $\Psi_n$. Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<\eta_k n^{(3^k-1)/(2k)-1}$, with $-\log \eta_k \sim c3^k$ and the same $c$.