Abstract
We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over \({\mathbb F}\) . 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.
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Ben-Eliezer, I., Hod, R. & Lovett, S. Random low-degree polynomials are hard to approximate. comput. complex. 21, 63–81 (2012). https://doi.org/10.1007/s00037-011-0020-6
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DOI: https://doi.org/10.1007/s00037-011-0020-6