On a Class of Ternary Inclusion-exclusion Polynomials

A ternary inclusion-exclusion polynomial is a polynomial of the form Q{p,q,r} = (zpqr − 1)(zp − 1)(zq − 1)(zr − 1) (zpq − 1)(zqr − 1)(zrp − 1)(z − 1) , where p, q, and r are integers ≥ 3 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 ≡ s (mod pq). More specifically, we consider the case where 1 ≤ 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 ≥ 1 and r ≡ ±s (mod pq), then A(p, q, r) ≤ s.