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
11 - Incidences in Rd
Published online by Cambridge University Press: 17 March 2022
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
Summary
In this chapter, we study general incidence bounds in R^d. As a warm-up, we first derive an incidence bound for curves in R^3. The main result of this chapter is a general point-variety incidence bound in R^d. This result relies on another polynomial partitioning theorem, for the case where the points are on a constant-degree variety. The proof of this partitioning theorem relies on Hilbert polynomials. In particular, we use Hilbert polynomials to derive a polynomial ham sandwich theorem for points on a variety.
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- Information
- Polynomial Methods and Incidence Theory , pp. 155 - 171Publisher: Cambridge University PressPrint publication year: 2022