Curves in $\mathbb{R}^4$ and two-rich points

TL;DR

This paper establishes a new upper bound on the number of two-rich points formed by low degree algebraic curves in four-dimensional space, under certain distribution constraints.

Contribution

It introduces a novel bound on two-rich points in $\

Findings

Maximum of $C_\ extepsilon n^{4/3+3\textepsilon}$ two-rich points

Bound on curves in low degree hypersurfaces and surfaces

Structure theorem for arrangements with many two-rich points

Abstract

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_\epsilon n^{4/3+3\epsilon}$ two-rich points, provided at most $n^{2/3+2\epsilon}$ curves lie in any low degree hypersurface and at most $n^{1/3+\epsilon}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.