Polynomial Methods and Incidence Theory

The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture the cap set conjecture Erds''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.