Finite fields
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Pseudorandom Bits for Polynomials
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Inverse conjecture for the gowers norm is false
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Unconditional pseudorandom generators for low degree polynomials
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The Sum of d Small-Bias Generators Fools Polynomials of Degree d
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Worst Case to Average Case Reductions for Polynomials
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
IEEE Transactions on Information Theory
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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. This significantly improves previous results for such polynomials. 2. For the case of degree four polynomials with high Gowers norm we show that (a subspace of constant co-dimension of) Fn 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 previously shown 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.