Frank de Zeeuw

Polynomials vanishing on Cartesian products: The Elekes–Szabó 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.

An improved point-line incidence bound over arbitrary fields

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F2, with m7/8<n<m8/7, determine at most O(m11/15n11/15) incidences (where, if F has positive characteristic p, we assume m−2n13≪p15). This improves on the previous best-known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Becks theorem.

Extensions of a result of Elekes and Rónyai

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 Ronyai 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-Ronyai Theorem to a less symmetric cartesian product. This leads to a proof of Purdys conjecture with significantly fewer points on one of the lines. We also extend the Elekes-Ronyai 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.

Few distinct distances implies no heavy lines or circles

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set

The Elekes–Szabó Theorem in four dimensions

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On the Number of Ordinary Conics

of n points determines o(n) distinct distances, then no line contains Ω(n7/8) points of

On Sets Defining Few Ordinary Circles

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