On a Class of Ternary Inclusion-Exclusion Polynomials

A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{{p,q,r}}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of $Q_{{p,q,r}}$ and $Q_{{p,q,s}}$, with $r\equiv s\pmod{pq}$. More specifically, we consider the case where $1\le s<\max(p,q)<r$, and obtain a recursive estimate for the function $A(p,q,r)$--the function that gives the maximum of the absolute values of the coefficients of $Q_{{p,q,r}}$. A simple corollary of our main result is the following absolute estimate. If $s\ge1$ and $r\equiv\pm s\pmod{pq}$, then $A(p,q,r)\le s$.