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- W3101476879 abstract "A point $x in mathbb{F}^n$ is a joint formed by a finite collection $mathfrak{L}$ of lines in $mathbb{F}^n$ if there exist at least $n$ lines in $mathfrak{L}$ through $x$ that span $mathbb{F}^n$. It is known that there are $lesssim_n |mathfrak{L}|^{frac{n}{n-1}}$ joints formed by $mathfrak{L}$. We say that a point $x in mathbb{F}^n$ is a multijoint formed by the finite collections $mathfrak{L}_1,ldots,mathfrak{L}_n$ of lines in $mathbb{F}^n$ if there exist at least $n$ lines through $x$, one from each collection, spanning $mathbb{F}^n$. We show that there are $lesssim_n (|mathfrak{L}_1|cdots |mathfrak{L}_n|)^{frac{1}{n-1}}$ such points for any field $mathbb{F}$ and $n=3$, as well as for $mathbb{F}=mathbb{R}$ and any $n geq 3$. Moreover, we say that a point $x in mathbb{F}^n$ is a generic joint formed by a finite collection $mathfrak{L}$ of lines in $mathbb{F}^n$ if each $n$ lines of $mathfrak{L}$ through $x$ form a joint there. We show that, for $mathbb{F}=mathbb{R}$ and any $n geq 3$, there are $lesssim_n frac{|mathfrak{L}|^{frac{n}{n-1}}}{k^{frac{n+1}{n-1}}}+frac{|mathfrak{L}|}{k}$ generic joints formed by $mathfrak{L}$, each lying in $sim k$ lines of $mathfrak{L}$. This result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in $mathbb{R}^3$ in cite{Guth_Katz_2010} by Guth and Katz. Finally, we generalise our results in $mathbb{R}^n$ to the case of multijoints and generic joints formed by real algebraic curves." @default.
- W3101476879 created "2020-11-23" @default.
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- W3101476879 date "2015-06-04" @default.
- W3101476879 modified "2023-09-25" @default.
- W3101476879 title "Incidence Bounds on Multijoints and Generic Joints" @default.
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- W3101476879 doi "https://doi.org/10.1007/s00454-015-9703-0" @default.
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