论文标题
计数平面立方曲线在有限磁场上具有规定数量的合理交叉点
Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points
论文作者
论文摘要
For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points.我们将此设置为有关某个投影芦苇磨损代码的重量枚举器的问题。证明的主要输入包括计数的分数曲线对，这些曲线确实具有共同的组件，计数未能对立方体强加独立条件的点的配置以及Macwilliams Theorem从编码理论中的变化。
For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a problem about a weight enumerator of a certain projective Reed-Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.
