= Polynomial phase
{title2=$e^{2\pi iP(x)}$}
A polynomial phase is a complex-valued <function> of the form $f(x)=e^{2\pi iP(x)}$, where $P$ is a real <polynomial>, or a suitably defined polynomial taking values in $\mathbb R/\mathbb Z$. Its absolute value is one. On a finite cyclic <group>, a typical well-defined example is $f(x)=e^{2\pi i a x^2/N}$ with integer $a$ and $x\in\mathbb Z/N\mathbb Z$.
The <multiplicative derivative> satisfies $f(x+h)\overline{f(x)}=e^{2\pi i(P(x+h)-P(x))}$. An ordinary additive difference lowers the degree of a nonconstant <polynomial> by at least one. Consequently, if the degree is less than $k$, every $k$-fold multiplicative difference is one, and the normalized <Gowers uniformity norm> $U^k$ equals one on a finite <abelian group>. This explains why polynomial phases serve as structured comparison functions in <higher-order Fourier analysis>.
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