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We investigate the structure of polynomials of degree four in many variablesover a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Taoproved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorlydistributed, then it is a function of a few polynomials of smaller degree. In2009, 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 polynomialsrequired is at most polynomial in $1/\delta$ and $f$ has a simple presentationas a sum of their products. We make a step towards showing that in fact thenumber of lower degree polynomials required is at most log-polynomial in$1/\delta$, with the same simple presentation of $f$. This result was aMaster's thesis supervised by T. Ziegler at the Hebrew University of Jerusalem,submitted in October 2018. A log-polynomial bound for polynomials of arbitrarydegree was recently proved independently by Milicevic and by Janzer.

Bias implies low rank for quartic polynomials