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 ∈ F q 0 [x, y] and any F q /F q 0 , the image of the map F 3 q → F 3 q given by (s, x, y) → (s, sx + f (x, y), sy + g(x, y)) has size at least 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) = 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.Keywords: Kakeya problem; image set on F q -points; Lang-Weil bound; reducibility of polynomials in several variables; number of irreducible components of a variety; indecomposable polynomials; linearized polynomials; permutation polynomials.

Factorization type probabilities of polynomials with prescribed coefficients over a finite field

“…Proof. The author has proven this as Lemma 19 in [6]. For convenience of the reader, we sketch the proof here as well.…”

A lower bound for the size of Kakeya sets with respect to hyperplanes in $\mathbb{F}_q^n$

“…This proof is a discrete version of the previously mentioned proof in [Dav71]. We were not able to trace it back in the literature, but to read an exposition of this proof see [Sla14].…”