Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem

TL;DR

This paper introduces new algorithms based on semi-infinite programming and branch and bound techniques to find integer Chebyshev polynomials, achieving the discovery of 16 new polynomials of high degrees.

Contribution

The paper develops novel computational methods that improve the search for integer Chebyshev polynomials, extending the known set of such polynomials to higher degrees.

Findings

Discovered 16 new integer Chebyshev polynomials of degrees 147 to 244.

Implemented algorithms that outperform previous methods for this problem.

Enhanced understanding of polynomial minimization over integers.

Abstract

We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.