I have a set of random variables $X_1,\ldots,X_n$, all Gaussian with mean 0 and variance 1, indepedent. Let $p(x_1,\ldots,x_n)$ be some polynomial that takes products and sums of $x_1,\ldots,x_n$.

What can be said about the concentration of measure of $p(X_1,\ldots,X_n)$ around $E[p(X_1,\ldots,X_n)]$?

If there were only two-order interactions, I think I would look around for concentration of measure for chi-squared random variables, but unfortunately the interaction can be of a higher degree.