Polynomial Methods and Incidence Theory
Published on 9. March 2022
978-1-108-96301-5 (ISBN)
Description
The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdos'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.
Reviews / Votes
'This book gives a very nice introduction to the areas of incidence geometry and the polynomial method ... 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.' Audie Warren, zbMATH OpenMore details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
College/higher education
Illustrations
Worked examples or Exercises
File size
7,97 MB
ISBN-13
978-1-108-96301-5 (9781108963015)
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Schweitzer Classification
Other editions
Additional editions
Adam Sheffer
Polynomial Methods and Incidence Theory
Book
03/2022
Cambridge University Press
€71.60
Shipment within 15-20 days
Person
Content
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.
