Generalized Gorshkov–Wirsing Polynomials and the Integer Chebyshev Problem

Abstract

The integer Chebyshev problem is the problem of finding an integer polynomial of degree n such that the supremum norm on [0, 1] is minimized. The most common technique used to find upper bounds is by explicit construction of an example. This is often (although not always) done by heavy computational use of the LLL algorithm and simplex method. One of the first methods developed to find lower bounds employed a sequence of polynomials known as the Gorshkov–Wirsing polynomials.

This paper studies properties of the Gorshkov–Wirsing polynomials. It is shown how to construct generalized Gorshkov–Wirsing polynomials on any interval [a, b], with . An extensive search for generalized Gorshkov–Wirsing polynomials is carried out for a large family of [a, b]. Using generalized Gorshkov–Wirsing polynomials, LLL, and the simplex method, upper and lower bounds for the integer Chebyshev constant on intervals other than [0, 1] are calculated. These methods are compared with other existing methods.

2000 AMS Subject Classification:

• Primary 11C08
• Secondary 30C10

Keywords:

• Chebyshev polynomials
• Diophantine approximation
• integers
• Transfinite diameter

ACKNOWLEDGMENTS

The author's research was supported in part by NSERC of Canada.

Notes

¹The precise pairs, as well as more complete tables (up to denominator 5) can be found at the author's home page (http://www.math.uwaterloo.ca/ kghare ).