On a Class of Ternary Inclusion-Exclusion Polynomials

2010/06/02 by Gennady Bachman, Pieter Moree, Bachman, Gennady +1
Mathematics · #11B83 #11C08 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B83 #msc:11C08

paper · pdf · doi:10.48550/arxiv.1006.0522

12 pages

arxiv created 2010/06/02 · arxiv updated 2010/06/04

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 ≥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\pmodpq. 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\pmodpq, then A(p,q,r)≤ s.

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