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arXiv · 2602.07727

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

Gennady Bachman
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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 $p$.

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Gennady Bachman. 2026-02-07. On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials. https://arxiv.org/abs/2602.07727

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