An algebraic geometry version of the Kakeya problem

TL;DR

This paper introduces an algebraic geometry approach to the Kakeya problem over finite fields, proposing a conjecture on the size of certain polynomial image sets and proving special cases using advanced algebraic techniques.

Contribution

It formulates a new algebraic geometry framework for the Kakeya problem and proves the conjecture in specific polynomial cases, advancing understanding of polynomial image set sizes.

Findings

Conjecture on the minimal size of polynomial image sets over finite fields.

Proof of the conjecture when f and g are univariate polynomials.

Partial proof under additional conditions for linearized polynomials.

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.