Coefficients of (inverse) unitary cyclotomic polynomials

Abstract

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials ${\Phi }_n^{\ast }(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m\ge 1$ we show that every integer occurs as a coefficient of ${\Phi }_{mn}^{\ast }(x)$ for some $n\ge 1$. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/{\Phi }_n^{\ast }(x),$ the inverse unitary cyclotomic polynomials.

12 pages, 4 tables, to appear in the Kodai Mathematical Journal