Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials

Abstract

The maximum gap g(f ) of a polynomial f is the maximum of the differences (gaps) between two consecutive exponents that appear in f . Let Φn and Ψn denote the n-th cyclotomic and n-th inverse cyclotomic polynomial, respectively. In this paper, we give several lower bounds for g(Φn) and g(Ψn), where n is the product of odd primes. We observe that they are very often exact. We also give an exact expression for g(Ψn) under a certain condition. Finally we conjecture an exact expression for g(Φn) under a certain condition.