  Classes of homogeneous polynomials between Banach spaces have been studied in
the last three decades from the perspective of the so-called ideal property: if
a polynomial P belongs to a class Q, then the composition u o P o v of P with
linear operators u and v belongs to Q as well. In an attempt to explore the
nonlinearity of the subject in a more consistent way, and taking into account
recent results in the field, in this paper we propose the study of classes of
homogeneous polynomials that are stable under the composition with homogeneous
polynomials. Some importante classes justify the study of the intermediate
concept of classes of polynomials Q such that if P belongs to Q, u is a linear
operator and Q is a homogeneous polynomial, then u o P o Q belongs to Q.
