On the Structure of Quintic Polynomials
international workshop and international workshop on approximation randomization and combinatorial optimization algorithms and techniques(2016)
摘要
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.
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