Maximum Gap in (Inverse) Cyclotomic Polynomial

TL;DR

This paper investigates the maximum gaps between consecutive exponents in cyclotomic and inverse cyclotomic polynomials for products of odd primes, providing exact formulas and bounds for specific cases.

Contribution

It derives exact expressions for the maximum gaps in cyclotomic and inverse cyclotomic polynomials for certain prime products, advancing understanding of their structure.

Findings

Exact formula for g(Φ_{p_1 p_2})

Exact expression for g(Ψ_{p_1 p_2 p_3}) under mild conditions

Bounds for g(Ψ_{p_1 p_2 p_3})

Abstract

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(\Phi_n)$ and $g(\Psi_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(\Phi_{p_1})$, $g(\Psi_{p_1})$ and $g(\Psi_{p_1p_2})$. Hence the simplest non-trivial cases are $g(\Phi_{p_1p_2})$ and $g(\Psi_{p_1p_2p_3})$. We provide an exact expression for $g(\Phi_{p_1p_2}).$ We also provide an exact expression for $g(\Psi_{p_1p_2p_3})$ 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 $g(\Psi_{p_1p_2p_3})$.