Bias implies low rank for quartic polynomials

TL;DR

This paper explores the structure of degree four polynomials over finite fields, showing that high bias implies a low-rank representation with a logarithmic bound on the number of lower-degree polynomials involved.

Contribution

It advances understanding by establishing a log-polynomial bound on the number of lower-degree polynomials needed to represent biased quartic polynomials, improving previous polynomial bounds.

Findings

High bias implies a simple low-rank structure for quartic polynomials.

The number of lower degree polynomials needed is at most log-polynomial in 1/δ.

The result aligns with recent independent proofs for arbitrary degree polynomials.

Abstract

We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly distributed, then it is a function of a few polynomials of smaller degree. In 2009, Haramaty and Shpilka found an effective bound for $f$ of degree four: If $bias\left(f\right)\geq\delta$, then the number of lower degree polynomials required is at most polynomial in $1/\delta$ and $f$ has a simple presentation as a sum of their products. We make a step towards showing that in fact the number of lower degree polynomials required is at most log-polynomial in $1/\delta$, with the same simple presentation of $f$. This result was a Master's thesis supervised by T. Ziegler at the Hebrew University of Jerusalem, submitted in October 2018. A log-polynomial bound for polynomials of arbitrary degree was recently proved independently by Milicevic and by Janzer.