Book contents
- Frontmatter
- Dedication
- Contents
- Introduction
- 1 Incidences and Classical Discrete Geometry
- 2 Basic Real Algebraic Geometry in R2
- 3 Polynomial Partitioning
- 4 Basic Real Algebraic Geometry in Rd
- 5 The Joints Problem and Degree Reduction
- 6 Polynomial Methods in Finite Fields
- 7 The Elekes–Sharir–Guth–Katz Framework
- 8 Constant-Degree Polynomial Partitioning and Incidences in C2
- 9 Lines in R3
- 10 Distinct Distances Variants
- 11 Incidences in Rd
- 12 Incidence Applications in Rd
- 13 Incidences in Spaces Over Finite Fields
- 14 Algebraic Families, Dimension Counting, and Ruled Surfaces
- Appendix Preliminaries
- References
- Index
6 - Polynomial Methods in Finite Fields
Published online by Cambridge University Press: 17 March 2022
- Frontmatter
- Dedication
- Contents
- Introduction
- 1 Incidences and Classical Discrete Geometry
- 2 Basic Real Algebraic Geometry in R2
- 3 Polynomial Partitioning
- 4 Basic Real Algebraic Geometry in Rd
- 5 The Joints Problem and Degree Reduction
- 6 Polynomial Methods in Finite Fields
- 7 The Elekes–Sharir–Guth–Katz Framework
- 8 Constant-Degree Polynomial Partitioning and Incidences in C2
- 9 Lines in R3
- 10 Distinct Distances Variants
- 11 Incidences in Rd
- 12 Incidence Applications in Rd
- 13 Incidences in Spaces Over Finite Fields
- 14 Algebraic Families, Dimension Counting, and Ruled Surfaces
- Appendix Preliminaries
- References
- Index
Summary
In this chapter, we use polynomial methods to study incidence-related problems in spaces over finite fields. We focus on two breakthroughs: A solution to the finite field Kakeya problem and the cap set problem. The proofs of these results are short, elegant, and require mostly elementary tools. In Chapter 13, we study point-line incidences in spaces over finite fields, which require more involved arguments.
This chapter contains a variety of other interesting problems and tools. We study the method of multiplicities, which improves the constant of the finite-field Kakeya theorem. To study the cap set problem, we use the slice rank technique. This technique is also used to obtain bounds for the 3-sunflower problem. As a warm-up towards the slice rank technique, we consider the Odd town problem and the two distance problem.
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- Polynomial Methods and Incidence Theory , pp. 76 - 94Publisher: Cambridge University PressPrint publication year: 2022