论文标题
有限场上的大多数平面曲线都没有阻止
Most plane curves over finite fields are not blocking
论文作者
论文摘要
平面曲线$ c \ subset \ mathbb {p}^2 $ of度数$ d $的$ d $称为\ emph {blocking}，如果每个$ \ mathbb {f} _q $ - line在飞机上遇到$ c $在某些$ \ mathbb {f} f} _q $ - 点上的$ c $。我们证明，当$ d $ $ o（1）$ $ d \ geq 2q-1 $和$ q \ to \ to \ infty $时，阻止曲线的比例为$ o（1）$。我们还表明，在较弱的条件$ d \ geq 3p $和$ d，q \ to \ infty $的情况下，平滑曲线的结论相同。此外，随机平面曲线平滑且封闭的两个事件在渐近独立。扩展了$ \ mathbb {f} _q $ - 随机多项式的$ \ mathbb {f} _q $的经典结果，我们发现在随机平面曲线和固定$ \ mathbb {f} _q $ line的$ \ mathbb {f} _q $ - 点的$ \ mathbb {f} _q $ - 点$ 1 $ poisson的限制分布。我们还提供了一个明确的公式，用于涉及统计数据的块曲线比例，该曲线涉及$ \ mathbb {f} _q $ - $ k $行中包含的$ k $ lines的$ k = 1、2，\ ldots，q^2+q+1 $。
A plane curve $C\subset\mathbb{P}^2$ of degree $d$ is called \emph{blocking} if every $\mathbb{F}_q$-line in the plane meets $C$ at some $\mathbb{F}_q$-point. We prove that the proportion of blocking curves among those of degree $d$ is $o(1)$ when $d\geq 2q-1$ and $q \to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\geq 3p$ and $d, q \to \infty$. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $\mathbb{F}_q$-roots of random polynomials, we find that the limiting distribution of the number of $\mathbb{F}_q$-points in the intersection of a random plane curve and a fixed $\mathbb{F}_q$-line is Poisson with mean $1$. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $\mathbb{F}_q$-points contained in a union of $k$ lines for $k=1, 2, \ldots, q^2+q+1$.
