Аn analogue of Kolmogorov’s theorem on superpositions of continuous functions for functional systems of polynomial and rational functions

Abstract

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.
Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.
The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
The well-known Kolmogorov theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition adjoins the completeness problem (see the formulation of this theorem below).
The purpose of this paper is the following problem: is there an analogue of Kolmogorov’s theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition for functional systems of polynomial functions and functional systems of rational functions?
It turns out that the problem posed (an analogue of Kolmogorov’s theorem) has a negative answer for functional systems of polynomial functions with natural and integer coefficients, and for functional systems of polynomial functions with rational and real coefficients and for functional systems of rational functions with rational and real coefficients - the answer is positive. These theorems are the main results of this paper.