The multivariate integer Chebyshev problem

TL;DR

This paper investigates the multivariate integer Chebyshev problem, focusing on polynomial minimization over complex sets, and extends classical bounds to higher dimensions, providing new theoretical insights.

Contribution

It introduces a multivariate analog of the Hilbert-Fekete upper bound for the integer Chebyshev constant, applicable to general sets and product sets.

Findings

Established a multivariate Hilbert-Fekete upper bound

Extended classical univariate results to multivariate cases

Provided bounds for specific sets like cubes and polydisks

Abstract

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in $\C^d.$ We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert-Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the Hilbert-Fekete result.