On the Structure of Cubic and Quartic Polynomials

TL;DR

This paper characterizes the structure of degree three and four polynomials with high bias or Gowers norm over prime fields, revealing canonical forms and subspace decompositions that extend and improve prior results.

Contribution

It provides a canonical representation for biased degree three and four polynomials and shows that quartic polynomials with high Gowers norm can be approximated by degree three polynomials on large subspaces.

Findings

Canonical form for biased degree three and four polynomials

Decomposition of quartic polynomials with high Gowers norm into degree three polynomials on subspaces

Improves understanding of polynomial structure over prime fields

Abstract

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.