Polynomial Methods and Incidence Theory

A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.

Adam Sheffer (Author)

9781108832496, Cambridge University Press

Hardback, published 24 March 2022

260 pages
23.5 x 15.7 x 2.2 cm, 0.54 kg

The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erd?s's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas.

Introduction
1. Incidences and classical discrete geometry
2. Basic real algebraic geometry in R^2
3. Polynomial partitioning
4. Basic real algebraic geometry in R^d
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 C^2
9. Lines in R^3
10. Distinct distances variants
11. Incidences in R^d
12. Incidence applications in R^d
13. Incidences in spaces over finite fields
14. Algebraic families, dimension counting, and ruled surfaces
Appendix. Preliminaries
References
Index.

Subject Areas: Geometry [PBM], Algebra [PBF], Discrete mathematics [PBD]