  We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$
(denoted $L_{1,k}$), of various well-studied classes of "structured"
$\mathbb{F}_2$-polynomials. This study is motivated by applications in
pseudorandomness, in particular recent results and conjectures due to
[CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at
level $k=2$) give unconditional pseudorandom generators.
  Our main structural results on Fourier growth are as follows:
  - We show that any symmetric degree-$d$ $\mathbb{F}_2$-polynomial $p$ has
$L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and this is tight for any constant $k$.
This quadratically strengthens an earlier bound that was implicit in [RSV13].
  - We show that any read-$\Delta$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has
$L_{1,k}(p) \le \Pr[p=1] \cdot (k \Delta d)^{O(k)}$.
  - We establish a composition theorem which gives $L_{1,k}$ bounds on disjoint
compositions of functions that are closed under restrictions and admit
$L_{1,k}$ bounds.
  Finally, we apply the above structural results to obtain new unconditional
pseudorandom generators and new correlation bounds for various classes of
$\mathbb{F}_2$-polynomials.
