This book is primarily a textbook introduction to various areas of discrete geometry. In each
area, it explains several key results and methods, in an accessible and concrete manner. It …

This paper surveys how the concept of crossing number, which used to be familiar only to a
limited group of specialists, emerges as a significant graph parameter. This paper has dual …

Let $ P:\F\times\F\to\F $ be a polynomial of bounded degree over a finite field $\F $ of large
characteristic. In this paper we establish the following dichotomy: either $ P $ is a moderate …

Abstract Let F∈ C [x, y, z] be a constant-degree polynomial, and let A, B, C⊂ C be finite sets
of size n. We show that F vanishes on at most O (n 11/6) points of the Cartesian product A× …

In this paper we characterize real bivariate polynomials which have a small range over large
Cartesian products. We show that for every constant-degree bivariate real polynomial f …

Geometric questions which involve Euclidean distances often lead to polynomial relations of
type F (x, y, z)= 0 for some F∈ ℝ [x, y, z]. Several problems of Combinatorial Geometry can …

We survey recent progress in the combinatorial analysis of incidences between points and
curves and in estimating the total combinatorial complexity of a set of faces in arrangements …

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian
product A× B× C with an algebraic surface {f= 0} over the reals. In particular, if A, B, C are …

We determine which quadratic polynomials in three variables are expanders over an
arbitrary field F F. More precisely, we prove that for a quadratic polynomial f∈ FF x, y, z …

Let S be a set of n points in R2 contained in an algebraic curve C of degree d. We prove that
the number of distinct distances determined by S is at least cdn4/3, unless C contains a line …