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2 results

  • 6 - Polynomial Methods in Finite Fields

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 76-94

  • Summary

    In this chapter, we use polynomial methods to study incidence-related problems in spaces over finite fields. We focus on two breakthroughs: A solution to the finite field Kakeya problem and the cap set problem. The proofs of these results are short, elegant, and require mostly elementary tools. In Chapter 13, we study point-line incidences in spaces over finite fields, which require more involved arguments.

    This chapter contains a variety of other interesting problems and tools. We study the method of multiplicities, which improves the constant of the finite-field Kakeya theorem. To study the cap set problem, we use the slice rank technique. This technique is also used to obtain bounds for the 3-sunflower problem. As a warm-up towards the slice rank technique, we consider the Odd town problem and the two distance problem.

Polynomial Methods and Incidence Theory
  • Adam Sheffer
  • Published online:
    17 March 2022
    Print publication:
    24 March 2022
  • 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.