论文标题
关于常规复合物中的线的发生率
On incidences of lines in regular complexes
论文作者
论文摘要
常规线性线复合物是空间中的三参数线，其plücker向量位于超平面中，与克莱因二次不相切。我们的主要结果是$ o（n^{1/2} m^{3/4} + m + n）$，用于复杂中的$ n $行之间的发病率数，而$ \ mathbb f^3 $中的$ n $ lition和$ m $点，其中$ \ m athbb f $是a field a field a field，a field a field，$ n \ n \ leq char（\ leq char（\ m m m m m m m马理）patitation。 Zahl最近观察到，来自两个不同的线复合体的双线线的双重成对发病率是$ \ Mathbb f^3 $的一组$ n $点的非零单距离问题。这暗示着新的bound $ o（n^{3/2}）$用于实现距离的数量，即$ \ mathbb f $，其中$ -1 $不是$ \ mathbb f $ analogue of erd \ h h os单距离$ \ \ m m i \ m m i \ m i \ m i \ m r^3 $中的正方形。在自然约束下，我们的发病率绑定产量较弱的$ o（n^{1.6}）$，该$在任何距离（包括零）中保持在任何$ \ mathbb f $上。
A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound $O(n^{1/2}m^{3/4} + m+n)$ for the number of incidences between $n$ lines in a complex and $m$ points in $\mathbb F^3$, where $\mathbb F$ is a field, and $n\leq char(\mathbb F)^{4/3}$ in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences of lines coming from two distinct line complexes account for the nonzero single distance problem for a set of $n$ points in $\mathbb F^3$. This implied the new bound $O(n^{3/2})$ for the number of realisations of the distance, which is a square, for $\mathbb F$, where $-1$ is not a square in the $\mathbb F$-analogue of the Erd\H os single distance problem in $\mathbb R^3$. Our incidence bound yields, under a natural constraint, a weaker bound $O(n^{1.6})$, which holds for any distance, including zero, over any $\mathbb F$.
