Maximum Gap in (Inverse) Cyclotomic Polynomial

2011/01/22 by Hong, Hoon, Lee, Eunjeong, Lee, Hyang-Sook +1 · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1101.4255

Abstract

Let g(f) denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial f. Let Φ_n denote the n-th cyclotomic polynomial and let Ψ_n denote the n-th inverse cyclotomic polynomial. In this note, we study g(Φ_n) and g(Ψ_n) where n is a product of odd primes, say p₁ < p₂ < p₃, etc. It is trivial to determine g(Φ_p1), g(Ψ_p1) and g(Ψ_p1p2). Hence the simplest non-trivial cases are g(Φ_p1p2) and g(Ψ_p1p2p3). We provide an exact expression for g(Φ_p1p2). We also provide an exact expression for g(Ψ_p1p2p3) under a mild condition. The condition is almost always satisfied (only finite exceptions for each p₁). We also provide a lower bound and an upper bound for g(Ψ_p1p2p3).

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• Gaps in binary cyclotomic polynomials

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