On a Class of Ternary Inclusion-Exclusion Polynomials

TL;DR

This paper investigates the coefficients of a class of ternary inclusion-exclusion polynomials, deriving recursive bounds and estimates for their maximum coefficient absolute values, extending understanding beyond cyclotomic polynomials.

Contribution

It introduces recursive estimates for the maximum coefficients of these polynomials when parameters satisfy certain modular conditions, generalizing previous results on cyclotomic polynomials.

Findings

Established bounds for maximum coefficients based on modular relations

Derived recursive estimates for coefficient magnitude

Provided explicit bounds when parameters satisfy specific congruences

Abstract

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 $A(p,q,r)\le s$.