## Abstract

In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field F, for any polynomial f on n variables with degree d ≤ log(n)/10, there exists a subspace of Fn with dimension Ω (d·n^{1/(d-1)}) on which f is constant. This result is shown to be tight. Stated differently, a degree d polynomial cannot compute an affine disperser for dimension smaller than Ω (d·n^{1/(d-1)}). Using a recursive argument, we obtain our second structural result, showing that any degree d polynomial f induces a partition of F^{n} to affine subspaces of dimension Ω (n^{1/(d-1)}!), such that f is constant on each part. We extend both structural results to more than one polynomial. We further prove an analog of the first structural result to sparse polynomials (with no restriction on the degree) and to functions that are close to low degree polynomials. We also consider the algorithmic aspect of the two structural results. Our structural results have various applications, two of which are: Dvir [11] introduced the notion of extractors for varieties, and gave explicit constructions of such extractors over large fields. We show that over any finite field any affine extractor is also an extractor for varieties with related parameters. Our reduction also holds for dispersers, and we conclude that Shaltiel's affine disperser [26] is a disperser for varieties over F_{2}. Ben-Sasson and Kopparty [6] proved that any degree 3 affine disperser over a prime field is also an affine extractor with related parameters. Using our structural results, and based on the work of Kaufman and Lovett [19] and Haramaty and Shpilka [17], we generalize this result to any constant degree.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 |

Editors | Naveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 680-709 |

Number of pages | 30 |

ISBN (Electronic) | 9783939897897 |

DOIs | |

State | Published - 1 Aug 2015 |

Externally published | Yes |

Event | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States Duration: 24 Aug 2015 → 26 Aug 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 40 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 |
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Country/Territory | United States |

City | Princeton |

Period | 24/08/15 → 26/08/15 |

## Keywords

- Affine dispersers
- Affine extractors
- Dispersers for varieties
- Extractors for varieties
- Low degree polynomials