Polynomials Vanishing On Cartesian Products: The Elekes-Szabo Theorem Revisited

Let F 2 C[x; y; z] be a constant-degree polynomial, and let A; B; C subset of 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 X B X C, unless F has a special group-related form. This improves a theorem of Elekes and Szab and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over R, and a similar statement holds when A; B; C have different sizes (with a more involved bound replacing O(n(11/6)). This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Extensions of a result of Elekes and Rónyai

Frank de Zeeuw

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and Rónyai proved that if the graph of a polynomial f(x, y) contains cn2 points of an n × n × n cartesian product in R3, then the polynomial has one of the forms f(x, y) = g(k(x) + l(y)) or f(x, y) = g(k(x)l(y)). They used this to prove a conjecture of Purdy which states that given two lines in R2 and n points on each line, if the number of distinct distances between pairs of points, one on each line, is at most cn, then the lines are parallel or orthogonal. We extend the Elekes-Rónyai Theorem to a less symmetric cartesian product. This leads to a proof of Purdy's conjecture with significantly fewer points on one of the lines. We also extend the Elekes-Rónyai Theorem to n × n × n × n cartesian products, again with an asymmetric version. We finish with a lower bound which shows that our result for asymmetric cartesian products in four dimensions is near-optimal. © 2013 Elsevier Inc.

Elsevier2013