  In this paper we study the structure of polynomials of degree three and four
that have high bias or high Gowers 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 Green and Tao and Kaufman and Lovett
for such polynomials.
  2. For the case of degree four polynomials with high Gowers 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 by Green and Tao and by Lovett,
Meshulam and Samorodnitsky that a quartic polynomial with a high Gowers 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.
