Another way of proving that something is nonzero is to prove that it is odd. One good example of that idea is the proof of Sperner's lemma
More generally, one can prove that something is nonzero by proving that it is nonzero mod p. That is the idea used in Chevalley-Warning theorem (one proves that the number of solutions is 0 mod p, then proves there is a trivial solution), and in the proof of Cauchy's theorem.
