Rutgers Discrete Mathematics Seminar

Title: On The Number of Ordinary Lines Determined By Sets in Complex Space

Speaker: Abdul Basit, Rutgers University

Date: Monday, February 6, 2017 2:00 pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ


Consider a set of n points in R^d. The classical theorem of Sylvester-Gallai says that, if the points are not all collinear then there must be a line through exactly two of the points. We call such a line an "ordinary line". In a recent result, Green and Tao were able to give optimal linear lower bounds (roughly n/2) on the number of ordinary lines determined n non-collinear points in R^d. In this talk we will consider the analog over the complex numbers. While the Sylvester-Gallai theorem as stated above is known to be false over the field of complex numbers, it was shown by Kelly that for a set of n points in C^d, if the points don’t all lie on a 2-dimensional plane then the points must determine an ordinary line. Using techniques developed for bounding the rank of design matrices, we will show that such a point set must determine at least 3n/2 ordinary lines, except in the trivial case of n−1 of the points being contained in a 2-dimensional plane.

Joint work with Z. Dvir, S. Saraf and C. Wolf