We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

翻译：本文提出并解决了热带代数框架下的离散最佳逼近问题，该代数体系涉及具有幂等加法的半环与半域。给定一组由未知函数（定义在幂等半域上）的输入输出构成的样本集合，问题目标是寻找该函数的热带普瑟多项式与有理函数最佳逼近。我们提出了一种新的求解方法，将多项式逼近问题归结为单侧含未知向量的热带线性向量方程的最佳近似解问题。推导出该单侧方程的最佳近似解，并以直接解析形式评估固有逼近误差。进一步将有理逼近问题转化为双侧含未知向量的方程最佳近似解，通过迭代交替算法获得其数值解。为阐明所发展的新技术，我们在实数半域（加法定义为取最大值，乘法定义为算术加法，即最大-加法代数）中求解示例逼近问题，对应分段线性函数的最佳切比雪夫逼近。