Construction de nombres extr\'emaux pour le probl\`eme de l'approximation simultan\'ee d'un nombre et de son carr\'e

TL;DR

This paper develops a general framework for constructing extremal numbers that simultaneously approximate a number and its square, extending to imaginary quadratic fields and rational function fields.

Contribution

It introduces a novel method for constructing extremal numbers in a broad algebraic setting, including imaginary quadratic and rational function fields.

Findings

Constructed new extremal numbers in general algebraic frameworks.

Extended approximation techniques to imaginary quadratic and rational function fields.

Provided a unified approach to simultaneous approximation problems.

Abstract

We consider the problem of simultaneous approximation to a number and to its square in a general framework that encompasses imaginary quadratic number fields and fields of rational functions in one variable. In this context, we construct new extremal numbers.