This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all mononials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial $x_1^2 x_2^2 x_3$ on the euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex.

翻译：本文关注第一类单变量切比雪夫多项式到多元情形的扩展，即在均匀范数下，寻找特定单项式的低次多项式最佳逼近。利用矩-SOS 层次结构，我们设计了一种基于半定规划的多功能计算方法，用于计算此类最佳逼近及其特征标识。将该方法应用于三个变量，我们获得了欧几里得球、单纯形和交叉多面体上所有不超过六次单项式的最佳逼近误差值。此外，受数值实验启发，我们推导出两个此前未解决情形下切比雪夫多项式的显式表达式：即欧几里得球上的单项式 $x_1^2 x_2^2 x_3$ 和单纯形上的单项式 $x_1^2 x_2 x_3$。