  We study random variables of the form $f(X)$, when $f$ is a degree $d$
polynomial, and $X$ is a random vector on $\mathbb{R}^{n}$, motivated towards a
deeper understanding of the covariance structure of $X^{\otimes d}$. For
applications, the main interest is to bound $\mathrm{Var}(f(X))$ from below,
assuming a suitable normalization on the coefficients of $f$. Our first result
applies when $X$ has independent coordinates, and we establish dimension-free
bounds. We also show that the assumption of independence can be relaxed and
that our bounds carry over to uniform measures on isotropic $L_{p}$ balls.
Moreover, in the case of the Euclidean ball, we provide an orthogonal
decomposition of $\mathrm{Cov}(X^{\otimes d})$. Finally, we utilize the
connection between anti-concentration and decay of Fourier coefficients to
prove a high-dimensional analogue of the van der Corput lemma, thus partially
answering a question posed by Carbery and Wright.
