# The Elekes-Rónyai-Szabó theory and its applications

## Micha Sharir, Tel Aviv University

## February 16, 2015

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 $A \times B \times C$?

This setup arises in many interesting problems in combinatorial
geometry,
including distinct distances between points on curves, distinct
distances
from three points, collinear triples of points on curves (the ‘orchard
problem’),
unit-area triangles, triple intersection points of families of
circles, and more.

This question has been studied by Elekes and Róonyai
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
quadratically many points of $A \times B \times C$, or else f must have one of the
special forms
$f(x,y) = h(p(x)+q(y))$ or $f(x,y) = h(p(x)q(y))$, for univariate
polynomials $p$, $q$, $h$.

In this talk I will survey recent progress on this problem, 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 $O(n^{11/6})$. Moreover,
the
results also hold over the complex field. This yields significantly
improved
bounds for many geometric problems, as listed above.

The proofs use techniques from algebra and algebraic geometry, which
are somewhat related
to the recent growing body of work on algebraic techniques for
incidences and distance
problems, inspired by Guth and Katz's seminal papers.

Joint work with Orit Raz, Jozsef Solymosi, and Frank de Zeeuw (and others).