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
5 - The Joints Problem and Degree Reduction
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
Before their seminal distinct distances paper, Guth and Katz wrote another paper that introduced a new polynomial method. In this chapter, we study one of the two problems that were resolved in that paper: the joints problem. The solution to this problem relies on a simple polynomial technique, which is based on polynomial interpolation. This is also a good warm-up for working in spaces of dimension larger than two.
We use the polynomial interpolation technique to study two additional problems. First, we study the sets in R^3 that are formed by the union of all lines that intersect three pairwise-skew lines. We then use the degree reduction technique to study polynomial interpolation of lines.
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- Polynomial Methods and Incidence Theory , pp. 66 - 75Publisher: Cambridge University PressPrint publication year: 2022