## Experimental Mathematics

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

### 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 $a, b ∈ \mathbb{Q}$. 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