  We prove a structural result for degree-$d$ polynomials. In particular, we
show that any degree-$d$ polynomial, $p$ can be approximated by another
polynomial, $p_0$, which can be decomposed as some function of polynomials
$q_1,...,q_m$ with $q_i$ normalized and $m=O_d(1)$, so that if $X$ is a
Gaussian random variable, the probability distribution on $(q_1(X),...,q_m(X))$
does not have too much mass in any small box.
  Using this result, we prove improved versions of a number of results about
polynomial threshold functions, including producing better pseudorandom
generators, obtaining a better invariance principle, and proving improved
bounds on noise sensitivity.
