Coefficients of (inverse) unitary cyclotomic polynomials

2019/11/05 by Gareth A. Jones, P. I. Kester, Jones, G. +11
Computer Science · Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1911.01749

openalex publication_date 2019/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials Φ_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≥ 1 we show that every integer occurs as a coefficient of Φ^*_mn(x) for some n≥ 1. Here n will typically have many different prime factors. We also consider similar questions for the polynomials (xⁿ-1)/Φ_n^*(x), the inverse unitary cyclotomic polynomials.

Related

• Unitary Cyclotomic Polynomials
• Coefficients of Unitary Cyclotomic Polynomials of Order Three
• Divisibility criteria and coefficient formulas for cyclotomic polynomials
• Some properties of coefficients of cyclotomic polynomials
• Coefficients and higher order derivatives of cyclotomic polynomials: old\n and new