= Quadratic phase
On $\mathbb F_p^n$, a classical <quadratic phase> is
$$
f(x)=e_p(q(x)),\qquad e_p(t)=e^{2\pi it/p},
$$
where $q$ is a <polynomial> of <polynomial degree> at most two. It may include linear and constant terms; a homogeneous <quadratic form> is a special case. For example, $q(x_1,x_2)=x_1^2+x_2+1$ gives a <quadratic phase> without being a homogeneous <quadratic form>.
Every third <multiplicative derivative> of this phase is one, so its <Gowers U3 norm> is one. A second <multiplicative derivative> is already constant in the base <point>; this explains why the cube average detects quadratic structure. In characteristic two, the full <inverse theorem for the Gowers U3 norm over a finite field> uses the broader nonclassical phase formulation, as stated there.
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