On a Fekete-Szeg\"o Theorem

TL;DR

This paper revisits a classical theorem linking capacities and algebraic integers, exploring the approximation of multiple complex numbers simultaneously by conjugate algebraic integers of fixed degree.

Contribution

It provides new insights into the classical Fekete-Szeg"o theorem and its applications to simultaneous approximation problems.

Findings

Established refined bounds for approximation of complex numbers.

Connected capacity theory with algebraic integer approximation.

Extended classical results to new contexts.

Abstract

We consider again a classical theorem relating capacities and algebraic integers and the question of the simultaneous approximation of $ n-1$ different complex numbers by conjugate algebraic integers of degree $n$.