  The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is
the minimum degree of a real polynomial $p$ that approximates $f$ pointwise:
$|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we
construct CNF and DNF formulas of polynomial size with approximate degree
$\Omega(n^{1-\delta}),$ essentially matching the trivial upper bound of $n.$
This improves polynomially on previous lower bounds and fully resolves the
approximate degree of constant-depth circuits ($\text{AC}^0$), a question that
has seen extensive research over the past 10 years. Previously, an
$\Omega(n^{1-\delta})$ lower bound was known only for $\text{AC}^0$ circuits of
depth that grows with $1/\delta$ (Bun and Thaler, FOCS 2017). Moreover, our CNF
and DNF formulas are the simplest possible in that they have constant width.
Our result holds even for one-sided approximation, and has the following
further consequences.
  (i) We essentially settle the communication complexity of $\text{AC}^0$
circuits in the bounded-error quantum model, $k$-party number-on-the-forehead
randomized model, and $k$-party number-on-the-forehead nondeterministic model:
we prove that for every $\delta>0$, these models require
$\Omega(n^{1-\delta})$, $\Omega(n/4^kk^2)^{1-\delta}$, and
$\Omega(n/4^kk^2)^{1-\delta}$, respectively, bits of communication even for
polynomial-size constant-width CNF formulas.
  (ii) In particular, we show that the multiparty communication class
$\text{coNP}_k$ can be separated essentially optimally from $\text{NP}_k$ and
$\text{BPP}_k$ by a particularly simple function, a polynomial-size
constant-width CNF.
  (iii) We give an essentially tight separation, of $O(1)$ versus
$\Omega(n^{1-\delta})$, for the one-sided versus two-sided approximate degree
of a function; and $O(1)$ versus $\Omega(n^{1-\delta})$ for the one-sided
approximate degree of a function $f$ versus its negation $\neg f$.
