Polynomials with integer coefficients and their zeros

TL;DR

This paper investigates polynomials with integer coefficients, focusing on their zeros, approximation properties, and coefficient growth, with implications for classical problems like the integer Chebyshev problem and algebraic number means.

Contribution

It provides new insights into the distribution of zeros and coefficient growth of integer coefficient polynomials, connecting these to approximation and algebraic number theory.

Findings

Distribution of zeros influences polynomial approximation

Coefficient growth relates to root location

Applications to classical integer polynomial problems

Abstract

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.