On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials

TL;DR

This paper investigates the coefficients' heights and diameters of ternary cyclotomic polynomials, proving that all heights within a certain range occur and providing explicit constructions with specific diameter values.

Contribution

It demonstrates that every integer height within a range occurs for some cyclotomic polynomial and explicitly constructs such polynomials with controlled diameters.

Findings

Every integer height from 1 to (p+1)/2 occurs for some cyclotomic polynomial.

Explicit constructions of primes q and r achieve specified heights.

Diameter of these polynomials is either 2h or 2h-1, depending on h modulo p.

Abstract

For the $n$th cyclotomic polynomial $\Phi_n$, let $A(n)$ denote the greatest absolute value of its coefficients, its height, and let $D(n)$ denote the difference between its largest and smallest coefficients, its diameter. We show that for any odd prime $p$ and an integer $h$ in the range $1\le h\le(p+1)/2$, there are arbitrarily large primes $q$ and $r$ such that $\Phi_{pqr}$ has the height $h$. This certainly answers the question of whether every natural number occurs as the height of some cyclotomic polynomial. Our construction specifies explicit choices of $q$ and $r$ with $A(pqr)=h$, and for these choices $D(pqr)$ has one of two values: it is either $2h$ or $2h-1$, depending on the congruence class of $h$ modulo $p$.