On the joints problem in finite fields

I think that Quilodrán's solution to the joints problem in $\mathbb{R}^n$ can be applied to the finite field case, to get the same bounds. This is the paper, "The joints problem in $\mathbb{R}^n$": Abstract: arXiv:0906.0555v3; PDF link.

I know I'm digging up an old thread, but I figured out how to extend Kaplan, Sharir and Shustin's proof (similar to Quilodran's proof) to finite fields last summer, then later realized that Dvir indicates the proof in a set of lecture notes.

Anyway, the trick is this: if $Q$ is a polynomial over $\mathbb{F}_q[x_1,...,x_n]$ whose gradient vanishes identically, then $Q$ is the $p$th power of some other polynomial $Q_1$ (where $q$ is a power of $p$). Since $Q_1$ is zero if and only if $Q_1^p$ is zero, if we're assuming that $Q$ is the minimum degree polynomial that vanishes on the set of lines forming our joints, we get a contradiction.

Dvir mentions how to do this in his lecture notes: http://www.cs.princeton.edu/~zdvir/teaching/incidence12/6.%20The%20polynomial%20method.pdf

I couldn't get Hasse derivatives to work, but it might be possible. I forgot what the trouble was with them.