Bias implies low rank for quartic 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 $f:{\mathbb{F}}^n\to \mathbb{F}$ 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 $bias\left(f\right)\ge \delta$, 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.