Cotype and absolutely summing homogeneous polynomials in Lp spaces

In this paper we lift to homogeneous polynomials and multilinear mappings a linear result due to Lindenstrauss and Pe{\l}czy\'{n}ski for absolutely summing mappings. We explore the notion of cotype to obtain stronger results and provide various examples of situations in which we have the space of absolutely summing polynomials different from the whole space. Among other consequences, these results enable us to obtain answers to some open questions about absolutely summing polynomials and multilinear mappings on $\mathcal{L}_{\infty}$ spaces.