On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials
Gennady Bachman
TL;DR
The paper advances the understanding of heights and diameters of ternary cyclotomic and inclusion-exclusion polynomials by constructing, for any odd prime $p$ and each $h$ with $1\le h\le(p+1)/2$, arbitrarily large primes $q,r$ such that $A(pqr)=h$, with explicit choices for $q$ and $r$. It develops a comprehensive framework using $Q_T$ and the auxiliary sums $S(Q,R;M)$ to reduce the height problem to a finite optimization over triples $(Q,R;M)$, and performs a detailed three-case analysis to determine $A^+(T)$ and $A^-(T)$, from which the height and diameter follow. The results yield constructive solubility of $A(pqr)=h$ for the entire range $1\le h\le(p+1)/2$ and reveal that $D(pqr)$ attains values in {2,3,p} and, more generally, can realize many even and odd diameters via explicit congruence choices; in particular $D(pqr)$ equals either $2h$ or $2h-1$ depending on $h\bmod p$. The methods also extend to inclusion-exclusion polynomials and rely on prime-distribution tools (e.g., Dirichlet progressions) and a detailed combinatorial analysis of representable integers relative to $qr$, $pr$, and $pq$.
Abstract
For the $n$th cyclotomic polynomial $Φ_n$, 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 $Φ_{pqr}$ 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 $p$.