  The goal of this paper is to estimate the total variation distance between
two general stochastic polynomials. As a consequence one obtains an invariance
principle for such polynomials. This generalizes known results concerning the
total variation distance between two multiple stochastic integrals on one hand,
and invariance principles in Kolmogorov distance for multi-linear stochastic
polynomials on the other hand. As an application we first discuss the
asymptotic behavior of U-statistics associated to polynomial kernels. Moreover
we also give an example of CLT associated to quadratic forms.
