On the Structure of Quintic Polynomials

TL;DR

This paper proves that degree five polynomials over finite fields with bias exhibit a specific algebraic structure and are constant on large affine subspaces, extending known results for degrees up to four.

Contribution

It establishes the structure of biased degree five polynomials and their constancy on large affine subspaces, extending prior work from degree four to five.

Findings

Degree five biased polynomials can be decomposed into structured components plus lower degree polynomials.

Such polynomials are constant on large affine subspaces of the domain.

The results hold for polynomials with degree less than the field size plus four.

Abstract

We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let $\mathbb{F}=\mathbb{F}_q$ be a prime field. [1.] Suppose $f:\mathbb{F}^n\rightarrow \mathbb{F}$ is a degree five polynomial with bias(f)=\delta. Then f can be written in the form $f= \sum_{i=1}^{c} G_i H_i + Q$, where $G_i$ and $H_i$s are nonconstant polynomials satisfying $deg(G_i)+deg(H_i)\leq 5$ and $Q$ is a degree $\leq 4$ polynomial. Moreover, $c=c(\delta)$ does not depend on $n$ and $q$. [2.] Suppose $f:\mathbb{F}^n\rightarrow \mathbb{F}$ is a degree five polynomial with $bias(f)=\delta$. Then there exists an $\Omega_\delta(n)$ dimensional affine subspace $V$ of $\mathbb{F}^n$ such that $f$ restricted to $V$ is a constant. Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension $\Omega(n)$. Item [2.] extends this to degree five polynomials. A corollary to Item [2.] is that any degree five affine disperser for dimension $k$ is also an affine extractor for dimension $O(k)$. We note that Item [2.] cannot hold for degrees six or higher. We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when $d<|\mathbb{F}|+4$. While the $d<|\mathbb{F}|+4$ assumption seems very restrictive, we note that prior to our work such structure theorems were only known for $d<|\mathbb{F}|$ by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et. al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n.