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 $f,g\in\F_{q_0}[x,y]$ 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.