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 $P:V\rightarrow k$ we define the bias $\tilde{b}_1(P)$ to be

$$\begin{aligned} \tilde{b}_1(P)=\frac{|\sum _{v\in V}\psi (P(v))|}{q^N} \end{aligned}$$

where $\psi :k\rightarrow {\mathbb {C}}^\star$ 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 $\tilde{b}_1(P)\ge c$ can be written as a sum $P=\sum _{i=1}^rQ_iR_i$ where $Q_i,R_i:V\rightarrow k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.