Dec. 3, 3:30pm, Wean 8220

Orit Raz, Tel Aviv University

Polynomials vanishing on Cartesian products

Orit Raz, Tel Aviv University

Polynomials vanishing on Cartesian products

Abstract:

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
striking results is that, for the special case where *F*(*x*,*y*,*z*) = *z*-*f*(*x*,*y*),
either *F* vanishes at *o*(*n ^{2}*) number of points of

In the talk I will discuss several recent results, in which the analysis is
greatly simplified, and the
bounds become sharp: If *F* does not have a special form, the number of
roots is at most *O*(*n ^{11/6}*). The
results also hold over the complex field.

This setup arises in various Erdős-type problems in combinatorial geometry,
and the result provides a
unified tool for their analysis. I will discuss an application of this kind
to the following problem: Given *n* points lying on a *d*-dimensional
algebraic variety in *R ^{D}*, show that there always exists a large subset

Based on joint works with Micha Sharir, Jozsef Solymosi, and Frank de Zeeuw