  In this work we derive multi-level concentration inequalities for polynomial
functions in independent random variables with a $\alpha$-sub-exponential tail
decay. A particularly interesting case is given by quadratic forms $f(X_1,
\ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type
inequalities with explicit dependence on various norms of the matrix $A$. A
consequence of these inequalities is a two-level concentration inequality for
quadratic forms in $\alpha$-sub-exponential random variables, such as quadratic
Poisson chaos.
  We provide various applications of these inequalities. Among these are
generalizations the results given by Rudelson-Vershynin from sub-Gaussian to
$\alpha$-sub-exponential random variables, i. e. concentration of the Euclidean
norm of the linear image of a random vector, small ball probability estimates
and concentration inequalities for the distance between a random vector and a
fixed subspace. Moreover, we obtain concentration inequalities for the excess
loss in a fixed design linear regression and the norm of a randomly projected
random vector.
