Counting integer polynomials with several roots of maximal modulus

TL;DR

This paper estimates the number of monic integer polynomials of degree n with bounded height that have exactly k roots of maximal modulus, including counts of reducible and irreducible cases, revealing asymptotic behaviors.

Contribution

It provides new bounds and counts for monic integer polynomials with specified roots of maximal modulus, including irreducible and reducible cases, for fixed degree and height.

Findings

Number of irreducible polynomials with all roots of equal modulus is about 2H for odd n.

For even n, there are more than H^{n/8} such polynomials.

Derived bounds relate polynomial height to root distribution.

Abstract

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ of such polynomials.