Maximum gap in (inverse) cyclotomic polynomial

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 p 1 < p 2 < p 3 , etc. It is trivial to determine g ( Φ p 1 ) , g ( Ψ p 1 ) and g ( Ψ p 1 p 2 ) . Hence the simplest non-trivial cases are g ( Φ p 1 p 2 ) and g ( Ψ p 1 p 2 p 3 ) . We provide an exact expression for g ( Φ p 1 p 2 ) . We also provide an exact expression for g ( Ψ p 1 p 2 p 3 ) under a mild condition. The condition is almost always satisfied (only finite exceptions for each p 1 ). We also provide a lower bound and an upper bound for g ( Ψ p 1 p 2 p 3 ) .