Alon's (or Alon and Tarsi's?) combinatorial nullstellensatz is a powerful algebraic tool with many applications in combinatorics and number theory. See [this][1], [this][2], [this][3] and [this mathoverflow question][4]. 

I am looking for good examples of results that were proved using combinatorial nullstellensatz (or its generalisation) but have no other known proof. 


  [1]: http://www.tau.ac.il/~nogaa/PDFS/null2.pdf
  [2]: http://arxiv.org/abs/1310.6482
  [3]: http://arxiv.org/abs/1008.2901
  [4]: http://mathoverflow.net/questions/43317/how-to-recognise-that-the-polynomial-method-might-work