Experimental Mathematics

Generalized Gorshkov–Wirsing Polynomials and the Integer Chebyshev Problem

Kevin G. Hare

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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 $a, b ∈ \mathbb{Q}$. 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.

Article information

Source
Experiment. Math., Volume 20, Issue 2 (2011), 189-200.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.em/1317924410

Mathematical Reviews number (MathSciNet)
MR2821390

Zentralblatt MATH identifier
1269.11029

Subjects
Primary: 11C08: Polynomials [See also 13F20]
Secondary: 30C10: Polynomials

Keywords
Chebyshev polynomials Diophantine approximation integers Transfinite diameter

Citation

Hare, Kevin G. Generalized Gorshkov–Wirsing Polynomials and the Integer Chebyshev Problem. Experiment. Math. 20 (2011), no. 2, 189--200. https://projecteuclid.org/euclid.em/1317924410


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