Random low-degree polynomials are hard to approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over 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 Theta(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.