Finite field Szemeredi-Trotter theorem with unequal number of points and lines

Currently, the only way to get non-trivial Szemeredi-Trotter bounds for medium-sized sets of points and lines in finite fields is to first reduce to a sum-product type problem, then prove a non-trivial sum-product estimate. The latter has a good chance of extending to asymmetric settings, but the former is still problematic there, basically because one is now only quasi-extremal to one trivial bound rather than to two, which allows for substantially more variability and non-rigidity in any putative counterexample to improvements over the trivial bounds. (For instance, in the Helfgott-Rudnev explicit version of this inequality at http://arxiv.org/abs/1001.1980 , they focus exclusively on the latter (in the symmetric setting) but don't improve the former beyond what Nets, Jean, and I did in http://arxiv.org/abs/math/0301343.

For very small sets (of logarithmic size or so), one can embed the configuration into the complex plane, at which point the complex Szemeredi-Trotter theorem becomes available; see this paper of Grosu, http://arxiv.org/abs/1303.2363 , for details. Unfortunately it is not clear how to push this method to larger sets.

Given that the Szemeredi-Trotter theorem over the reals (and complexes) can be proven by the polynomial method (see e.g. my survey on that method at http://arxiv.org/abs/1310.6482), and that the polynomial method has had other notable successes in finite fields, it is very natural to suspect that the polynomial method should be usable to deduce non-trivial Szemeredi-Trotter or sum-product theorems in finite fields. It is thus a bit frustrating that no such proof is currently known (the key obstacle being the lack of a substitute for the ham sandwich theorem for the finite field setting).

I believe one can deduce an improvement of the form $$I(P,L) \leq |P| |L|^{1/2-\epsilon}$$ from the symmetric case $|P|=|L|$ by using the following argument of Pudlak (see Corollary 2.5, there): Assume that $|P| > |L|$ and select a random subset $P'$ of $P$ of size $L$. The expected number of incidences will be $I(P,L) \frac{|L|}{|P|}$. Let $P'$ denote a set of points at or above the expectation. Now applying the symmetric result to the symmetric incidence problem with $P'$ and $L$, should give $$I(P,L) \frac{|L|}{|P|} \leq I(P',L) \leq |L|^{3/2-\epsilon'}. $$

In the case of large sets, there is also a result of Le Anh Vinh, which states that:

$$I(P,L) \leq q^{-1} |P| |L| + q^{1/2} |P|^{1/2} |L|^{1/2}. $$

Here, $q$ is the order of the finite field. Note, however, that this is worse than trivial when $|P|\times|L|$ is smaller than $q$.

In case anyone stumbles across this thread, there is now a nice paper by Sophie Stevens (https://arxiv.org/abs/1609.06284) which improves upon the Cauchy-Schwarz bound for all ranges of $m$ and $n$ with $m^{1/2} \leq n \leq m^2$.