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On Wirsing's problem in small exact degree

Johannes Schleischitz

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 Győry on a problem of Szegedy.

On Wirsing's problem in small exact degree

Abstract

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly . This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to . 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 Győry on a problem of Szegedy.

Paper Structure

This paper contains 18 sections, 18 theorems, 133 equations.

Table of Contents

  1. Introduction and main results
  2. Wirsing's problem in exact degree
  3. Related new results in exact degree
  4. Irreducibility criteria for integer polynomials
  5. Auxiliary results
  6. A large set of polynomials small at $\xi$
  7. Proof of Theorems \ref{['t1']}, \ref{['t2']} for $n\leq 5$
  8. Case 1
  9. Case 2
  10. Case 3
  11. Case 4
  12. Proof of Theorem \ref{['t2']} for $n\leq 5$
  13. Preparations for $n=6$ and $n=7$
  14. Proof of Theorems \ref{['t1']}, \ref{['t2']} for $n=6$
  15. Proof of Theorems \ref{['t1']}, \ref{['t2']} for $n=7$
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $1\leq n\leq 7$ an integer and any transcendental real number $\xi$ we have

Theorems & Definitions (37)

  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Theorem 1.2
  • proof : Deduction of Theorem \ref{['H']} from Theorem \ref{['t1']}
  • Definition 3
  • Theorem 1.3
  • Definition 4
  • Remark 1
  • Theorem 1.4
  • ...and 27 more