Some polynomial versions of cotype and applications

We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st. and cotype, and spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on ℓ1ℓ1-multipliers of vector-valued Dirichlet series. Finally we introduce cotype with respect to indexing sets, an idea that includes our previous definitions.