On four-rich points defined by pencils

2025/08/26 by Kokkinos, Michalis, Warren, Audie
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.19061

Abstract

In this paper we study the number of four-rich points defined by pencils of certain algebraic objects. Our main result concerns the number of four-rich points defined by four sheaves of planes; under certain non-degeneracy conditions, we prove that four sheaves of n planes in \mathbb P³ determine at most O(n^8/3) four-rich points. We prove this using the four dimensional Elekes-Szabó theorem. Using the same method, we prove an upper bound on the number of four-rich points determined by four sets of concentric spheres in \mathbb C³. Furthermore, using the same technique with the 3-d Elekes-Szabó theorem, one can prove upper bounds on four-rich points determined by various configurations of lines/circles in the plane \mathbb C²; we give one such example, involving two pencils of lines and two pencils of concentric circles in \mathbb C².

Citations

• Improved Bounds for Pencils of Lines
• A survey of Elekes-Rónyai-type problems
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

Related

• Improved Bounds for Pencils of Lines
• On the number of rich lines in truly high dimensional sets
• On rich points and incidences with restricted sets of lines in 3-space
• Covering points by hyperplanes and related problems
• The Elekes-Szabó Theorem in four dimensions