Polynomial methods and incidence theory. (English) Zbl 1505.51001
Cambridge Studies in Advanced Mathematics 197. Cambridge: Cambridge University Press (ISBN 978-1-108-83249-6/hbk; 978-1-108-95998-8/ebook). xvi, 246 p. (2022).
This book gives a very nice introduction to the areas of incidence geometry and the polynomial method, together with further chapters which take a deeper look at some famous problems in discrete geometry, the most notable being the distinct distances problem, essentially solved by Guth and Katz using the polynomial method. Other famous problems solved with the polynomial method, including the joints problem and the finite field Kakeya problem, are studied.
The first four chapters serve as a gentle introduction, including classical incidence geometry, polynomial partitioning, and some basic algebraic geometry. Chapter 5 then proves the joints problem and considers some other applications of the method used. In Chapter 6, polynomial methods over finite fields are studied, including the Kakeya problem. Chapter 7 then builds the setting for the distinct distances problem – namely the Elekes-Sharir framework. Chapters 8 and 9 finish the proof, by setting up the polynomial partitioning over the complex numbers (and using it to prove the complex Szemerédi-Trotter theorem) in Chapter 8, and then applying this method in Chapter 9 to complete the proof. Chapter 10 then looks at variants of the distinct distances problem. Chapters 11 and 12 move to more generality, by considering incidences and their applications in the space \(\mathbb R^d\) instead of \(\mathbb R^2\) or \(\mathbb R^3\). Chapter 13 looks at the fast evolving and difficult field of incidence theory over finite fields (this includes an introduction to the projective plane), and the final Chapter 14 discusses some advances techniques from algebraic geometry.
The text is written for someone with no knowledge of the subject – merely an interest in combonatorial geometry and some basic algebra would be enough. This makes the book a strong choice for a masters or PhD student interested in the area. Since this area of mathematics is still rather young, the book contains many open problems – this helps to bring the reader to the front of research. Furthermore, each chapter is followed by a generous amount of exercises.
The first four chapters serve as a gentle introduction, including classical incidence geometry, polynomial partitioning, and some basic algebraic geometry. Chapter 5 then proves the joints problem and considers some other applications of the method used. In Chapter 6, polynomial methods over finite fields are studied, including the Kakeya problem. Chapter 7 then builds the setting for the distinct distances problem – namely the Elekes-Sharir framework. Chapters 8 and 9 finish the proof, by setting up the polynomial partitioning over the complex numbers (and using it to prove the complex Szemerédi-Trotter theorem) in Chapter 8, and then applying this method in Chapter 9 to complete the proof. Chapter 10 then looks at variants of the distinct distances problem. Chapters 11 and 12 move to more generality, by considering incidences and their applications in the space \(\mathbb R^d\) instead of \(\mathbb R^2\) or \(\mathbb R^3\). Chapter 13 looks at the fast evolving and difficult field of incidence theory over finite fields (this includes an introduction to the projective plane), and the final Chapter 14 discusses some advances techniques from algebraic geometry.
The text is written for someone with no knowledge of the subject – merely an interest in combonatorial geometry and some basic algebra would be enough. This makes the book a strong choice for a masters or PhD student interested in the area. Since this area of mathematics is still rather young, the book contains many open problems – this helps to bring the reader to the front of research. Furthermore, each chapter is followed by a generous amount of exercises.
Reviewer: Audie Warren (Linz)
MSC:
| 51-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry |
| 26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |
| 05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |
| 52C10 | Erdős problems and related topics of discrete geometry |
| 11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |
| 05B25 | Combinatorial aspects of finite geometries |
| 11B25 | Arithmetic progressions |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 51E22 | Linear codes and caps in Galois spaces |
