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Polynomial ideals from a nonlinear viewpoint

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.