Algebraic Solution of Tropical Best Approximation Problems

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