Monic polynomials in $Z[x]$ with roots in the unit disc

TL;DR

This paper investigates monic polynomials with integer coefficients whose roots lie in the unit disc, characterizing their structure, and determining the count of such polynomials for small degrees, building on Kronecker's classical results.

Contribution

It provides a canonical form for Kronecker polynomials and explicitly determines the sequence of their counts for small degrees, extending Kronecker's theorem.

Findings

Number of Kronecker polynomials of degree n is finite.

Roots of these polynomials lie on the boundary of the unit disc.

Such polynomials are products of cyclotomic polynomials.

Abstract

This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let $k(n)$ denote the number of Kronecker polynomials of degree $n$. We describe a canonical form for such polynomials and use it to determine the sequence $k(n)$, for small values of $n$. The first step is to show that the number of Kronecker polynomials of degree $n$ is finite. This fact is included in the following theorem due to Kronecker. The theorem actually gives more: the non-zero roots of such polynomials are on the boundary of the unit disc. We use this fact later on to show that these polynomials are essentially products of cyclotomic polynomials.