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arXiv · 1012.3897

On the height of cyclotomic polynomials

Bartlomiej Bzdega
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Abstract

Let $A_n$ denote the height of cyclotomic polynomial $Φ_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le ε_kϕ(n)^{k^{-1}2^{k-1}-1}$ with $-\logε_k\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $Ψ_n$. 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 $B_n<η_k n^{(3^k-1)/(2k)-1}$, with $-\log η_k \sim c3^k$ and the same $c$.

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Bartlomiej Bzdega. 2010-12-17. On the height of cyclotomic polynomials. https://arxiv.org/abs/1012.3897

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