Coefficients of (inverse) unitary cyclotomic polynomials

TL;DR

This paper investigates the coefficients of unitary cyclotomic polynomials and their inverses, revealing that all integers can appear as coefficients for certain polynomial indices, especially when n has multiple prime factors.

Contribution

It introduces the study of coefficients of unitary cyclotomic polynomials with multiple prime factors using numerical semigroups and inclusion-exclusion polynomials, showing universality of integer coefficients.

Findings

Every integer appears as a coefficient of rac{mn} for some n.

Analysis of coefficients when n has two or three prime factors.

Extension of results to inverse unitary cyclotomic polynomials.

Abstract

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. 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 $\Phi^*_{mn}(x)$ for some $n\ge 1$. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/\Phi_n^*(x),$ the inverse unitary cyclotomic polynomials.