# Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm.

A nonclassical polynomial of degree $d$ is a function $\tilde{f}: \mathbb{F}^k_p \rightarrow \mathbb{R}/\mathbb{Z}$ such that taking $d+1$ directional derivatives of $\tilde{f}$ always yields 0.

If one considers a natural embedding of $\mathbb{F}_p$ into $\mathbb{R}/\mathbb{Z}$ where $k \in \mathbb{F}_p$ is identified with $k/p$, then every polynomial $f: \mathbb{F}^k_p \rightarrow \mathbb{F}_p$ of degree at most $d$ has a corresponding to a nonclassical polynomial $\tilde{f}$ of degree at most $d$.

I am wondering to what extent the range of the nonclassical polynomial needs to be $\mathbb{R}/\mathbb{Z}$ as opposed to, for example, an arbitrary group. Though of course, the group must contain an element of order $p$ for embedding classical polynomials.

It seems plausible to me that one needs $\mathbb{R}/\mathbb{Z}$ to achieve the interesting properties desired; it also seems plausible that considering an arbitrary group yields not much interesting beyond the case $\mathbb{R}/\mathbb{Z}$. Or, maybe there is a rich and different theory.

Is there possibly a simple example which perhaps provides intuition one way or another?

See this blog post for more background on nonclassical polynomials.

• There is the more general theory of polynomial maps between two arbitrary groups, pioneered by Leibman; see e.g. Appendix B of this paper of Green, Ziegler, and myself: arxiv.org/abs/1009.3998 Dec 16, 2015 at 15:31