Curves in ${\mathbb{R}}^4$ and 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.

20 pages, 0 figures. v2: fixed an error in Section 5.2

References in corpus (2)

• Polynomial partitioning for a set of varieties
• A Szemeredi-Trotter type theorem in ${\mathbb{R}}^4$

Cited by in corpus (1)

• Breaking the 3/2 barrier for unit distances in three dimensions