Here are some old examples of applying polynomial method to incidence problems:

http://www.ams.org/journals/proc/1998-126-11/S0002-9939-98-04653-X/S0002-9939-98-04653-X.pdf

http://www.sciencedirect.com/science/article/pii/S0195669883710115

http://www.tau.ac.il/~nogaa/PDFS/null2.pdf

http://www.sciencedirect.com/science/article/pii/0097316578900134