Closed classes in the functional system of polynomials with real coefficients

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; this problem consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
In our paper we consider the functional system of polynomials with real coefficients endowed with the superposition operation for this system we study the problem of closed classes (structure, basis, number of finite and infinite closed classes).
Importance of the problem of closed classes is ensured by the fact that completeness problem can frequently be solved with the help of (maximal) closed classes.
The main results concerning the functional system of polynomials with real coefficients presented in our paper are the following:

• all finite closed classes are described explicitly;
• the number of finite closed classes, infinite closed classes and all closed classes is found;
• the problem of bases of closed classes is studied, namely, it is established that there exist closed classes with a finite basis, there exist closed classes with an infinite basis, and there exist closed classes without a basis; explicit examples of the corresponding closed classes are given;
• the number of closed classes with a finite basis, the number of closed classes with an infinite basis and the number of closed classes without a basis are established.