On the structure of cubic and quartic polynomials

Elad Haramaty*, Amir Shpilka

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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. 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) double-struck Fn can be partitioned to subspaces of dimension Ω(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.

Original languageEnglish
Title of host publicationSTOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing
Pages331-340
Number of pages10
DOIs
StatePublished - 2010
Externally publishedYes
Event42nd ACM Symposium on Theory of Computing, STOC 2010 - Cambridge, MA, United States
Duration: 5 Jun 20108 Jun 2010

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference42nd ACM Symposium on Theory of Computing, STOC 2010
Country/TerritoryUnited States
CityCambridge, MA
Period5/06/108/06/10

Keywords

  • Gowers norm
  • cubic polynomials
  • quartic polynomials

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