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

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 $p$.

💡 Research Summary

The paper investigates two fundamental invariants of cyclotomic polynomials Φₙ: the height A(n), defined as the maximum absolute value of its coefficients, and the diameter D(n), defined as the difference between the largest and smallest coefficients. While the behavior of these invariants is completely understood for cyclotomic polynomials of order k < 3 (they are “flat”, i.e., A(n)=1), the case k ≥ 3, especially the ternary case (three distinct odd prime factors), has remained largely open. In particular, it was unknown whether every natural number occurs as the height of some cyclotomic polynomial.

The authors introduce the ternary inclusion‑exclusion polynomial

 Q_{p,q,r}(x) = (x^{pqr}−1)(x^{p}−1)(x^{q}−1)(x^{r}−1) /