On a functional system of polynomials with rational coefficients

This article investigates the completeness problem for polynomial functions with rational coefficients, as well as problems of a functional nature generated by its solution, namely: studying the structure of closed and precomplete classes, the problem of bases of complete systems. In particular, it was proved that 1) the system of functions is complete if and only if it is not wholly contained in any precomplete class; 2) the cardinality of the set of all precomplete classes is equal to the continuum; 3) there is a complete system that does not have a basis.