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Two Structural Results for Low Degree Polynomials and Applications

Authors Gil Cohen, Avishay Tal

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Gil Cohen
Avishay Tal

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Gil Cohen and Avishay Tal. Two Structural Results for Low Degree Polynomials and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 680-709, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)


In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field F, for any polynomial f on n variables with degree d > log(n)/10, there exists a subspace of F^n with dimension at least d n^(1/(d-1)) on which f is constant. This result is shown to be tight. Stated differently, a degree d polynomial cannot compute an affine disperser for dimension smaller than the stated dimension. Using a recursive argument, we obtain our second structural result, showing that any degree d polynomial f induces a partition of F^n to affine subspaces of dimension n^(1/(d-1)!), such that f is constant on each part. We extend both structural results to more than one polynomial. We further prove an analog of the first structural result to sparse polynomials (with no restriction on the degree) and to functions that are close to low degree polynomials. We also consider the algorithmic aspect of the two structural results. Our structural results have various applications, two of which are: * Dvir [CC 2012] introduced the notion of extractors for varieties, and gave explicit constructions of such extractors over large fields. We show that over any finite field any affine extractor is also an extractor for varieties with related parameters. Our reduction also holds for dispersers, and we conclude that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over the binary field. * Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine disperser over a prime field is also an affine extractor with related parameters. Using our structural results, and based on the work of Kaufman and Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this result to any constant degree.
  • low degree polynomials
  • affine extractors
  • affine dispersers
  • extractors for varieties
  • dispersers for varieties


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