3 years ago

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

Larry Guth, Joshua Zahl

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.

Publisher URL: http://arxiv.org/abs/1512.05648

DOI: arXiv:1512.05648v2

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