关于定义几个普通圆的集合
On sets defining few ordinary circles
摘要：
平面中$n$点的集合$P$的普通圆被定义为恰好包含$P$的三个点的圆。我们表明，如果$P$不包含在一条线或一个圆圈，那么$P$跨越至少$\frac{1}{4}n^2 - O(n)$普通的圆。此外，我们确定的确切的最小数目的普通圆的所有足够大的$n$和描述所有的点集接近这个最小值。我们还考虑了果园问题的圆变体。我们证明了$P$最多跨越$\frac{1}{24}n^3 - O(n^2)$圆，恰好通过$P$的四个点。在这里，我们确定了所有足够大的$n$的确切的最大值和极值配置。这些结果是基于以下结构定理。如果$n$根据$K$足够大，并且$P$是一组至多跨越$Kn^2$普通圆的$n$点，则除了$P$的$O(K)$点之外，所有$n$点都位于次数至多为4的代数曲线上。我们的证明依赖于最近的结果绿色和陶普通线，结合循环反演和一些经典的结果，关于代数曲线。
英文摘要：
An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large $n$ and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that $P$ spans at most $\frac{1}{24}n^3 - O(n^2)$ circles passing through exactly four points of $P$. Here we determine the exact maximum and the extremal configurations for all sufficiently large $n$. These results are based on the following structure theorem. If $n$ is sufficiently large depending on $K$, and $P$ is a set of $n$ points spanning at most $Kn^2$ ordinary circles, then all but $O(K)$ points of $P$ lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.
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