Semialgebraic complexity of functions

In this paper we study the rate of the best approximation of a given function by semialgebraic functions of a prescribed “combinatorial complexity”. We call this rate a “Semialgebraic Complexity” of the approximated function. By the classical Approximation Theory, the rate of a polynomial approximation is determined by the regularity of the approximated function (the number of its continuous derivatives, the domain of analyticity, etc.). In contrast, semialgebraic complexity (being always bounded from above in terms of regularity) may be small for functions not regular in the usual sense. We give various natural examples of functions of low semialgebraic complexity, including maxima of smooth families, compositions, series of a special form, etc. We show that certain important characteristics of the functions, in particular, the geometry of their critical values (Morse-Sard Theorem) are determined by their semialgebraic complexity, and not by their regularity.