Random low degree polynomials are hard to approximate

Abstract

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over double-struck 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 Θ (n). 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. © 2009 Springer.