Random low-degree polynomials are hard to approximate

Ido Ben-Eliezer, Rani Hod, Shachar Lovett

Research output: Contribution to journalArticlepeer-review

Abstract

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials F 2. We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d - 1, for all degrees d up to Θ (n). That is, a random degree d 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.

Original languageEnglish
Pages (from-to)63-81
Number of pages19
JournalComputational Complexity
Volume21
Issue number1
DOIs
StatePublished - Mar 2012

Keywords

  • Random polynomials
  • Reed-Muller codes

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