On the bias of cubic polynomials

Abstract

Let $V$ be a vector space over a finite field $k={\mathbb{F}}_q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac{|{\sum }_{v\in V}\psi (P(V))|}{q^n}$$ where $\psi :k\to {\mathbb{C}}^{\star }$ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any $d\ge 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\ge c$ can be written as a sum $P={\sum }_{i=1}^r{Q}_i{R}_i$ where $Q_i,R_i:V\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.