On Wirsing's problem in small exact degree

TL;DR

This paper advances the understanding of Wirsing's approximation problem for real algebraic numbers of small degrees, providing improved bounds and new irreducibility criteria for polynomials.

Contribution

It improves bounds for degrees up to 7 in Wirsing's problem and introduces new irreducibility criteria for polynomial combinations, enhancing previous results.

Findings

Improved bounds for degrees up to 7 in Wirsing's approximation problem.

New irreducibility criteria for integral linear combinations of coprime polynomials.

Enhanced results for cubic polynomials related to Szegedy's problem.

Abstract

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we obtain results regarding small values of polynomials and approximation to a real number by algebraic integers in small prescribed degree. The main ingredient are irreducibility criteria for integral linear combinations of coprime integer polynomials. Moreover, for cubic polynomials these criteria improve results of Gy\H{o}ry on a problem of Szegedy.