Geometric incidences via the polynomial method

by · Jul 23, 2016 · 194 views ·

I will mention four recent breakthrough results in incidence geometry: Finite field Kakeya problem. (Dvir 2008) If AA is a subset of FnqFqn and contains a line in every direction, then |A|>cqn|A|>cqn, where cc depends on nn only. Joints problem. (Guth-Katz 2010) Any NN lines in the space determine at most O(N3/2)O(N3/2) joints (a joint is a point incident to three non-coplanar lines). Erdos distinct distance problem. (Guth-Katz 2015) Any NN points in the plane have at least cN/logNcN/logN distinct pairwise distances. Cap-set problem. (Ellenberg and Gijswijt, using a lemma of Croot, Lev and Pach. 2016+) This is a huge improvement on the bound for the size of a cap(-set) in affine space over the field GF(3)GF(3) of three elements. The common feature of the proofs of the four results is the smart use of a polynomial where the zero-set of the polynomial vanishes on a pointset determined by the problem. As a further illustration I will present a new application of the method, bounding the number of tangencies between algebraic curves in the plane. (Joint work with Jordan Ellenberg and Josh Zahl)

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