On the Structure of Cubic and Quartic Polynomials

In this paper we study the structure of polynomials of degree three and four that have high bias or high Cowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes the corresponding results from the theory of quadratic forms. It also significantly improves the results of [GT07] and [KL08] for such polynomials. 2. For the case of degree four polynomials with high Cowers norm we show that (a subspace of co-dimension O(1) of) F-n can be partitioned to subspaces of dimension Omega(n) such that on each of the subspaces the polynomial is equal to some degree three polynomial. It was shown in [GT07] and [LMS08] that a quartic polynomial with a high Cowers norm is not necessarily correlated with any cubic polynomial. Our result shows that a slightly weaker statement does hold. The proof is based on finding a structure in the space of partial derivatives of the underlying polynomial.