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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.33
URN: urn:nbn:de:0030-drops-66569
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6656/
Hatami, Pooya
On the Structure of Quintic Polynomials
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 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.
BibTeX - Entry
@InProceedings{hatami:LIPIcs:2016:6656,
author = {Pooya Hatami},
title = {{On the Structure of Quintic Polynomials}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-018-7},
ISSN = {1868-8969},
year = {2016},
volume = {60},
editor = {Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6656},
URN = {urn:nbn:de:0030-drops-66569},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.33},
annote = {Keywords: Higher-order Fourier analysis, Structure Theorem, Polynomials, Regularity lemmas}
}
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Keywords: |
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Higher-order Fourier analysis, Structure Theorem, Polynomials, Regularity lemmas |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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06.09.2016 |
