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El. knyga: Polynomial Methods and Incidence Theory

(Bernard M. Baruch College, City University of New York)
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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.

This is a detailed introduction to the new polynomial methods responsible for numerous major mathematical breakthroughs in the past decade. It requires a minimal background and includes many examples, warm-up proofs, and exercises, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front.

Recenzijos

'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 Open

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A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.
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 ElekesSharirGuthKatz
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.
Adam Sheffer is Mathematics Professor at CUNY's Baruch College and the CUNY Graduate Center. Previously, he was a postdoctoral researcher at the California Institute of Technology. Sheffer's research work is focused on polynomial methods, discrete geometry, and additive combinatorics.
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