An algebraic geometry version of the Kakeya problem

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials f,g∈Fq0[x,y]f,g∈Fq0[x,y] and any Fq/Fq0Fq/Fq0, the image of the map Fq3→Fq3 given by (s,x,y)↦(s,sx+f(x,y),sy+g(x,y))(s,x,y)↦(s,sx+f(x,y),sy+g(x,y)) has size at least q34−O(q5/2) and prove the special case when f=f(x),g=g(y)f=f(x),g=g(y). We also prove it in the case f=f(y),g=g(x)f=f(y),g=g(x) under the additional assumption f′(0)g′(0)≠0f′(0)g′(0)≠0 when f,gf,g are both affine polynomials. 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.