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On the Structure of Quintic Polynomials

Author Pooya Hatami

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Pooya Hatami

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Pooya Hatami. On the Structure of Quintic Polynomials. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 33:1-33:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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 F=F_q be a prime field. Suppose f:F^n to F is a degree five polynomial with bias(f)=delta. We prove the following two structural properties for such f. 1. We have f= sum_{i=1}^{c} G_i H_i + Q, where G_i and H_is are nonconstant polynomials satisfying deg(G_i)+deg(H_i)<= 5 and Q is a degree <5 polynomial. Moreover, c does not depend on n. 2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_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<|\F|+4. While the d<|F|+4 assumption seems very restrictive, we note that prior to our work such structure theorems were only known for d<|\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.
  • Higher-order Fourier analysis
  • Structure Theorem
  • Polynomials
  • Regularity lemmas


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