{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9073","name":"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.\r\n\r\n1. 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.\r\n\r\n2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_V is a constant.\r\n\r\nCohen 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.\r\n\r\nWe 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.","keywords":["Higher-order Fourier analysis","Structure Theorem","Polynomials","Regularity lemmas"],"author":{"@type":"Person","name":"Hatami, Pooya","givenName":"Pooya","familyName":"Hatami"},"position":33,"pageStart":"33:1","pageEnd":"33:18","dateCreated":"2016-09-06","datePublished":"2016-09-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Hatami, Pooya","givenName":"Pooya","familyName":"Hatami"},"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2016.33","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1145\/2488608.2488662","http:\/\/dx.doi.org\/10.1137\/1.9781611973730.125","http:\/\/arxiv.org\/abs\/1506.02047","http:\/\/dx.doi.org\/10.1145\/2746539.2746543","http:\/\/dx.doi.org\/10.1109\/FOCS.2008.17","http:\/\/dx.doi.org\/10.4086\/toc.2011.v007a009","http:\/\/dx.doi.org\/10.1017\/S030500410700093X","http:\/\/dx.doi.org\/10.1007\/s00026-011-0124-3"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6263","volumeNumber":60,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2016)","dateCreated":"2016-09-06","datePublished":"2016-09-06","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Umans, Chris","givenName":"Chris","familyName":"Umans"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9073","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6263"}}}