多关节计数
Counting multijoints
摘要：
令$\mathfrak{L}_1$、$\mathfrak{L}_2$、$\mathfrak{L}_3$分别是$L_1$、$L_2$、$L_3$的有限集合，$\mathbb{R}^3$中的线，以及$J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)$由它们形成的多关节的集合，即点$x \in \mathbb{R}^3$的集合，对于所有$i=1,2,3$，每个点位于至少一条线$l_i \in \mathfrak{L}_i$中，使得$l_1$、$l_2$和$l_3$的方向跨越$\mathbb{R}^3$。我们在这里证明了$|J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)|\lesssim (L_1L_2L_3)^{1/2}$，我们将我们的结果推广到多关节形成的真正的代数曲线在$\mathbb{R}^3$的一致有界的程度，以及曲线在$\mathbb{R}^3$参数化的真正的一元多项式的一致有界的程度。多关节问题是关节问题的一个变体，也是端点多线性Kakeya问题的离散模拟。
英文摘要：
Let $\mathfrak{L}_1$, $\mathfrak{L}_2$, $\mathfrak{L}_3$ be finite collections of $L_1$, $L_2$, $L_3$, respectively, lines in $\mathbb{R}^3$, and $J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)$ the set of multijoints formed by them, i.e. the set of points $x \in \mathbb{R}^3$, each of which lies in at least one line $l_i \in \mathfrak{L}_i$, for all $i=1,2,3$, such that the directions of $l_1$, $l_2$ and $l_3$ span $\mathbb{R}^3$. We prove here that $|J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)|\lesssim (L_1L_2L_3)^{1/2}$, and we extend our results to multijoints formed by real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as by curves in $\mathbb{R}^3$ 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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