# On the bias of cubic polynomials

## TL;DR

This paper investigates the bias of cubic polynomials over finite fields, establishing a modified converse statement to a known decomposition result for degree 3 polynomials.

## Contribution

It proves a modified version of the converse statement for the bias of cubic polynomials, extending previous results on polynomial decomposition over finite fields.

## Key findings

- Bias of cubic polynomials can be characterized by their decompositions.
- A modified converse statement holds for degree 3 polynomials.
- The results connect polynomial bias with algebraic structure.

## 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 $d\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\geq c$ can be written as a sum $P=\sum _{i=1}^rQ_iR_i$ 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 $d=3$.