Combinatorics seminar: Orit Raz

Mon, 18/12/201711:00-12:30
Eilat Hal at IIAS

Title: Polynomials vanishing on Cartesian products
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.