Clones of polynomials over infinite fields (Q1603066)

The main result of the paper is a description of the subsets of a polynomial ring in several variables over a field of characteristic \(0\) which are stable under compositions and cyclic and other natural actions on the variables. In particular, it is shown that the lattice of such subsets, interpolating between the linear forms and all polynomials without constant terms, is isomorphic to the lattice of all additive submonoids of \(\mathbb{N}\). Also, there is no such subset strictly between the sets of all linear polynomials and all polynomials.