A Note on Heights of Cyclotomic Polynomials

TL;DR

The paper addresses whether every positive integer occurs as the height $A(n)$ of a cyclotomic polynomial, focusing on the ternary case $n=pqr$ with distinct odd primes. It develops an approach via ternary inclusion–exclusion polynomials $Q_{\{p,q,r\}}(x)$ and a congruence-transfer mechanism: if $r\equiv s\pmod{pq}$, then $A(p,q,r)$ equals $A(p,q,s)$ with at most a unit jump, enabling controlled height realization. A key parametric lemma shows that for $p$ odd and suitable $q,r$, one can force $A(p,q,r)=(p+1)/2$, and thus produce $A(pqr)=h$ or $h+1$ for a target height $h$ by appropriate ordering of parameters. The main result (Theorem 1) proves that for every $h$, there exist primes $p,q,r$ such that $A(pqr)=h$ or $h+1$, and a corollary (Corollary 1) constructs a sparse prime set $\mathcal{P}$ with $P(x)<\log x$ supporting such triples for all $h$, highlighting an elementary pathway to realize all heights without heavy prime-distribution assumptions. Overall, the work reinforces inclusion–exclusion polynomials as a flexible framework for height realization in cyclotomic contexts and suggests further exploration of height sequences arising from ternary constructions.

Abstract

We show that for any positive integer $h$, either $h$ or $h+1$ is a height of some cyclotomic polynomial $Φ_n$, where $n$ is a product of three distinct primes.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2
• Theorem 1
• Corollary 1
• Conjecture
• Lemma 3
• Lemma 4
• proof : Proof of Theorem 1
• Lemma 5
• proof