  Decoupling inequalities disentangle complex dependence structures of random
objects so that they can be analyzed by means of standard tools from the theory
of independent random variables. We study decoupling inequalities for
vector-valued homogeneous polynomials evaluated at random variables. We focus
on providing geometric conditions ensuring decoupling inequalities with good
constants depending only exponentially on the degree of the polynomial.
Assuming the Banach space has finite cotype we achieve this for classical
decoupling inequalities that compare the polynomials with their associated
multilinear operators. Under stronger geometric assumptions on the involved
Banach spaces, we also obtain decoupling inequalities between random
polynomials and fully independent random sums of their coefficients. Finally,
we present decoupling inequalities where in the multilinear operator just two
independent copies of the random vector are involved (one repeated $m-1$
times).
