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 monomials 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：本文研究将第一类一元切比雪夫多项式推广至多元情形，其核心在于寻找相对于一致范数下对特定单项式的最佳低次多项式逼近。利用矩-平方和（Moment-SOS）层次结构，我们设计了一种基于半定规划的通用计算流程，用于求解此类最佳逼近多项式及其相关特征量。将该方法应用于三元情形，我们得到了欧几里得球体、单纯形和交叉多面体上所有六次及以下单项式的最佳逼近误差值。此外，受数值实验启发，我们在两个先前未解决的案例中获得了切比雪夫多项式的显式表达式：即欧几里得球体上单项式 $x_1^2 x_2^2 x_3$ 的最佳逼近多项式，以及单纯形上单项式 $x_1^2 x_2 x_3$ 的最佳逼近多项式。