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
4 - Basic Real Algebraic Geometry 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
This chapter consists of basic real algebraic geometry. In Chapter 2 we studied basic properties of curves in the plane. We now generalize these properties to arbitrary varieties in any real space and introduce additional properties.
We begin the chapter by briefly discussing polynomial ideals. We then study the dimension of a real variety, connected components, irreducible components, tangent spaces, singular points, regular points, and other properties. We discuss why there is no one well-behaved definition for the degree of a real variety. We also study the projection of real varieties and partitioning polynomials in higher dimensions.
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- Polynomial Methods and Incidence Theory , pp. 51 - 65Publisher: Cambridge University PressPrint publication year: 2022