On the height of cyclotomic polynomials

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 $A_n\le {\epsilon }_k\varphi (n{)}^{k^{-1}{2}^{k-1}-1}$ with $-\log {\epsilon }_k\sim c{2}^k$, $c>0$. 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 $-\log {\eta }_k\sim c{3}^k$ and the same $c$.

11 pages

References in corpus (1)

• A proof of the Corrected Beiter conjecture

Cited by in corpus (1)

• Maximum Gap in (Inverse) Cyclotomic Polynomial