Positive density of integer polynomials with some prescribed properties

TL;DR

This paper demonstrates that integer polynomials with specific root properties occur with positive density, including those with roots near prescribed points and certain Galois group structures, highlighting their abundance.

Contribution

It establishes the positive density of integer polynomials with prescribed root configurations and Galois groups, extending understanding of their distribution.

Findings

Most integer polynomials have 1 or 2 roots with maximal modulus.

A positive density of polynomials of degree n can have roots near any symmetric set of points.

Polynomials with Galois group S_n and roots close to given points are abundant.

Abstract

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer $n$ and any set of $n$ distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree $n$, height at most $H$ and Galois group $S_n$ whose roots are close to the given $n$ points.