45 citations · 145 across the 13 of their papers we have counts for
13 papers
Three-variable expanding polynomials and higher-dimensional distinct distances
Thang Pham, Le Anh Vinh, Frank de Zeeuw
We determine which quadratic polynomials in three variables are expanders over an arbitrary field . More precisely, we prove that for a quadratic polynomial $f\in \math…
On sets defining few ordinary circles
Aaron Lin, Mehdi Makhul, Hossein Nassajian Mojarrad +3
An ordinary circle of a set of points in the plane is defined as a circle that contains exactly three points of . We show that if is not contained in a line or a cir…
A survey of Elekes-Rónyai-type problems
Frank de Zeeuw
We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial has a large image…
Distinct distances between points and lines
Micha Sharir, Shakhar Smorodinsky, Claudiu Valculescu +1
We show that for points and lines in the real plane, the number of distinct distances between the points and the lines is , as long as $m^{1/2}\le n\le m…
On the number of ordinary conics
Thomas Boys, Claudiu Valculescu, Frank de Zeeuw
We prove a lower bound on the number of ordinary conics determined by a finite point set in . An ordinary conic for a subset of is a conic that is…
Schwartz-Zippel bounds for two-dimensional products
Hossein Nassajian Mojarrad, Thang Pham, Claudiu Valculescu +1
We prove bounds on intersections of algebraic varieties in with Cartesian products of finite sets from , and we point out connections with several clas…
Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited
Orit E. Raz, Micha Sharir, Frank de Zeeuw
Let be a constant-degree polynomial,and let be finite sets of size . We show that vanishes on at most points of…
Incidence bounds for complex algebraic curves on Cartesian products
József Solymosi, Frank de Zeeuw
We prove bounds on the number of incidences between a set of algebraic curves in and a Cartesian product with finite sets . Simila…
Bisector energy and few distinct distances
Ben Lund, Adam Sheffer, Frank de Zeeuw
We introduce the bisector energy of an -point set in , defined as the number of quadruples from such that and determine the same perpen…
Few distinct distances implies no heavy lines or circles
Adam Sheffer, Joshua Zahl, Frank de Zeeuw
We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distance…
Distinct distances on algebraic curves in the plane
János Pach, Frank de Zeeuw
Let be a set of points in the real plane contained in an algebraic curve of degree . We prove that the number of distinct distances determined by is at least $c_…
Extensions of a result of Elekes and Rónyai
Ryan Schwartz, József Solymosi, 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 th…
On a question of Erdos and Ulam
Jozsef Solymosi, Frank de Zeeuw
Ulam asked in 1945 if there is an everywhere dense \emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\H os conjectured that if a set $…