Random Low Degree Polynomials are Hard to Approximate

  • Authors:

  • Ido Ben-Eliezer;Rani Hod;Shachar Lovett

  • Affiliations:

  • Tel Aviv University,;Tel Aviv University,;Weizmann Institute of Science,

  • Venue:
  • APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
  • Year:
  • 2009

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Abstract

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over ${\mathbb{F}_{2}}$. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d , for all degrees d up to $\Theta\left(n\right)$. That is, a random degree d + 1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed---Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed---Muller codes.