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An algebraic geometry version of the Kakeya problem

Kaloyan Slavov

Abstract

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any $\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by $(s,x,y)\mapsto (s,sx+f(x,y),sy+g(x,y))$ has size at least $\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when $f=f(x), g=g(y).$ We also prove it in the case $f=f(y), g=g(x)$ under the additional assumption $f'(0)g'(0)\neq 0$ when $f,g$ are both linearized. Our approach is based on a combination of Cauchy--Schwarz and Lang--Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials.

An algebraic geometry version of the Kakeya problem

Abstract

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials and any , the image of the map given by has size at least and prove the special case when We also prove it in the case under the additional assumption when are both linearized. Our approach is based on a combination of Cauchy--Schwarz and Lang--Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials.

Paper Structure

This paper contains 12 sections, 14 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Definition of a Kakeya variety
  3. Some technical preparations
  4. Kakeya variety over a base field
  5. An explicit description
  6. Our approach
  7. The main idea
  8. Indecomposability of certain polynomials
  9. Main results
  10. The case of separated variables
  11. The case of mixed variables
  12. Open questions

Key Result

Proposition 2

Assume that $L(t_1,t_2)=L(t_1)$ and $M(t_1,t_2)=M(t_2)$ depend only on the first or second variable, respectively. Then Conjecture main_conj holds true.

Theorems & Definitions (39)

  • Conjecture 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Remark 7
  • Example 8
  • ...and 29 more