The polynomial method in combinatorial incidence geometry relies crucially on linear algebra to locate a non-trivial polynomial of controlled degree that vanishes at a specified set of points. A good example of the method in action is Dvir's proof of the finite field Kakeya conjecture, see e.g. http://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/ .
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