论文标题
三维曲线的发生率
Incidences with curves in three dimensions
论文作者
论文摘要
我们研究涉及点和曲线$ r^3 $的发病率问题。格斯（Guth）和卡兹（Katz）率先进行的当前（实际上仅是可行的）方法需要各种工具，即代数几何形状，最值得注意的是（i）多项式分配技术，以及（ii）对代数表面的研究，这些方面由线条或在较新的研究中，由algebraic curves统一的较新的研究统治。通过利用和完善这些工具，我们在$ r^3 $中获得了新的和改进的范围，以解决点曲线发病率问题。 从古斯（Guth）和卡兹（Katz）在点和线条上的工作开始，已经考虑了此类事件。我们的结果基于Guth和Zahl关于由曲线双重统治的表面的工作，对以前的大多数结果进行了巨大的概括。我们重建了点和线的界限，并以某些重要的方式改进了涉及点和圈子的最新界限（在Sharir，Sheffer和Zahl中），以及点和任意恒定的恒定度代数曲线（在Sharir，Sherir，Sheffer和Solomon中）。尽管在这些情况下，界限尚不清楚（并且强烈怀疑）要紧紧，但在某种意义上，考虑到当前的知识状态，我们的界限是可以通过这种方法获得的最好的界限。 作为我们的点曲线发病率的应用，我们表明，由$ r^3 $的一组$ n $点跨越的三角形数量，与给定的三角形类似为$ O（n^{15/7}）$，这改善了Agarwal等人的界限。我们的结果也与Guth等人的研究有关（正在进行的工作），最近在Sharir，Solomon和Zlydenko中应用于三个维度的相关发病率问题。
We study incidence problems involving points and curves in $R^3$. The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz, requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies, by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for point-curve incidence problems in $R^3$. Incidences of this kind have been considered in several previous studies, starting with Guth and Katz's work on points and lines. Our results, which are based on the work of Guth and Zahl concerning surfaces that are doubly ruled by curves, provide a grand generalization of most of the previous results. We reconstruct the bound for points and lines, and improve, in certain significant ways, recent bounds involving points and circles (in Sharir, Sheffer and Zahl), and points and arbitrary constant-degree algebraic curves (in Sharir, Sheffer and Solomon). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge. As an application of our point-curve incidence bound, we show that the number of triangles spanned by a set of $n$ points in $R^3$ and similar to a given triangle is $O(n^{15/7})$, which improves the bound of Agarwal et al. Our results are also related to a study by Guth et al.~(work in progress), and have been recently applied in Sharir, Solomon and Zlydenko to related incidence problems in three dimensions.
