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Here are some more examples of the polynomial method that may help in understanding when it can be used: - The blocking number of an affine space by Brouwer and Schrijver, JCT (A) (1978). - On the size of a blocking set in PG(2,p) by Blokhuis, Combinatorica (1994). - An easier proof of the maximal arcs conjecture by Simeon Ball and Aart Blokhuis, Proc. AMS (1998). And here's a book by Peter Sziklai on this topic: cs.elte.hu/~sziklai/polynom/poly08feb.pdf
