论文标题
近两个自由度的点和曲线之间的发生率
Incidences between points and curves with almost two degrees of freedom
论文作者
论文摘要
我们在三个维度上研究要点和代数曲线之间的事件，取自一个家庭$ c $的曲线，这些曲线具有近两个自由度，这意味着，每一条曲线在$ o（1）$上相交，对于任何一对点$ p $，$ q $，$ o（1）$ o（1）$ c $ cub yaver $ c $ cult $ $ $ $ $ $ p $通过两个$ f（p，q）= 0 $，对于某些多项式$ f $。 我们研究了两个特定实例，一个涉及$ r^3 $的单位圆，通过某个固定点（所谓的锚定单位圆圈），另一个涉及指向点（点和方向）和飞机上的圆圈之间的相对点；如果该点位于圆上，并且方向是切线方向，则指向点是切线的。 Ellenberg等人的举动转变。将这些切线映射到三个维度的点和曲线之间的事件。在这两种情况下，$ r^3 $中的曲线都具有近两个自由度。 我们表明，$ m $点和$ n $锚定的单位圆之间的发生率数量为$ r^3 $，以及$ m $指导点和飞机上$ n $ nutionary Circles之间的切线数为$ O（m^{3/5} n^{3/5} {3/5}+m+n）$。 我们在$ r^3 $中具有近两个自由度的更多一般曲线家庭，得出了类似的发病率，并提供了几个额外的术语。 这些证据遵循基于多项式分配的标准技术，但是面临一个新的问题，涉及一个由各自的曲线家族统治的表面，以及在双重3D空间中的表面，这些空间由相应定义的双曲线的各自家族统治。 我们获得的一般界限为$ O（m^{3/5} n^{3/5}+m+n）$加上其他术语，这些术语取决于数量或双曲线可以位于无限规定的表面上。
We study incidences between points and algebraic curves in three dimensions, taken from a family $C$ of curves that have almost two degrees of freedom, meaning that every pair of curves intersect in $O(1)$ points, for any pair of points $p$, $q$, there are only $O(1)$ curves of $C$ that pass through both points, and a pair $p$, $q$ of points admit a curve of $C$ that passes through both of them iff $F(p,q)=0$ for some polynomial $F$. We study two specific instances, one involving unit circles in $R^3$ that pass through some fixed point (so called anchored unit circles), and the other involving tangencies between directed points (points and directions) and circles in the plane; a directed point is tangent to a circle if the point lies on the circle and the direction is the tangent direction. A lifting transformation of Ellenberg et al. maps these tangencies to incidences between points and curves in three dimensions. In both instances the curves in $R^3$ have almost two degrees of freedom. We show that the number of incidences between $m$ points and $n$ anchored unit circles in $R^3$, as well as the number of tangencies between $m$ directed points and $n$ arbitrary circles in the plane, is $O(m^{3/5}n^{3/5}+m+n)$. We derive a similar incidence bound, with a few additional terms, for more general families of curves in $R^3$ with almost two degrees of freedom. The proofs follow standard techniques, based on polynomial partitioning, but face a novel issue involving surfaces that are infinitely ruled by the respective family of curves, as well as surfaces in a dual 3D space that are infinitely ruled by the respective family of suitably defined dual curves. The general bound that we obtain is $O(m^{3/5}n^{3/5}+m+n)$ plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.
