Counting multijoints

2014/01/24 by Iliopoulou, Marina · 1 citation
#Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1401.6392

Abstract

Let \mathfrakL₁, \mathfrakL₂, \mathfrakL₃ be finite collections of L₁, L₂, L₃, respectively, lines in ℝ³, and J(\mathfrakL₁, \mathfrakL₂,\mathfrakL₃) the set of multijoints formed by them, i.e. the set of points x ∈ ℝ³, each of which lies in at least one line l_i ∈ \mathfrakL_i, for all i=1,2,3, such that the directions of l₁, l₂ and l₃ span ℝ³. We prove here that |J(\mathfrakL₁, \mathfrakL₂,\mathfrakL₃)|\lesssim (L₁L₂L₃)^1/2, and we extend our results to multijoints formed by real algebraic curves in ℝ³ of uniformly bounded degree, as well as by curves in ℝ³ parametrised by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, as well as a discrete analogue of the endpoint multilinear Kakeya problem.

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