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Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds

Authors Abhishek Bhowmick, Shachar Lovett

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Abhishek Bhowmick
Shachar Lovett

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Abhishek Bhowmick and Shachar Lovett. Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 72-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.CCC.2015.72

Abstract

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

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Keywords
  • nonclassical polynomials
  • polynomials
  • lower bounds
  • barrier

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