Orit Raz is a Postdoctoral Fellow in the Department of Mathematics at the University of British Columbia in Vancouver. She received her Ph.D. from Tel-Aviv University, under the supervision of Micha Sharir. After graduating, she became a Member at the Institute for Advanced Study, in Princeton, New Jersey. She left in 2017 to start her Postdoctoral Fellowship at UBC.

Talk

Polynomials vanishing on Cartesian products

May 21, 2018

3:30 p.m.

LeConte 412

Let F(x,y,z) be a real trivariate polynomial of constant degree, and let A,B,C be three sets of real numbers, each of size n. How many roots can F have on A x B x C? This question has been studied by Elekes and Rónyai and then by Elekes and Szabó about 15 years ago. One of their results is that either F vanishes at o(n^2) number of points of A x B x C, or else the surface {F=0} must have a certain special form. In the talk I will discuss several recent results, in which the analysis is greatly simplified, and the bounds become sharp: If {F=0} does not have the special form, the number of roots is at most O(n^{11/6}). This setup arises in various Erdös-type problems in extremal combinatorial geometry, and the result mentioned above provides a unified tool for their analysis. If time allows, I will discuss some applications of this kind.