 Polynomial phase

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 $f(x)=e^{2\pi ia{x}^2/N}$ 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.