多项式方法与关联理论

由于引入了新的多项式方法，在过去的十年里，在有限域Kakeya猜想、帽集猜想、Erdős的不同距离问题、关节问题以及其他问题上取得了许多重大的数学突破。在加法组合学、离散几何等各种问题上也取得了重大进展。本书详细介绍了这些新的多项式方法及其应用，重点介绍了关联理论。根据作者自己的教学经验，本文只需要一点背景知识，让研究生和高年级本科生掌握一个活跃而令人兴奋的研究前沿。这些技巧是逐步详细介绍的，包括许多例子、热身练习和练习。附录提供了基本结果和想法的快速提醒。
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, Erdős’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.

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