论文标题
椭圆曲线的果园在有限田
Orchards in elliptic curves over finite fields
论文作者
论文摘要
考虑一组飞机上的$ n $点。 $ n $点中完全包含$ 3 $的线称为$ 3 $ -RICH线。经典的果园问题要求在飞机上使用$ n $点的配置，以最大化$ 3 $ - 富含$ -RICH的线路。在本说明中，使用椭圆形曲线的小组法上的有限磁场，我们展示了果园的几个（无限的）组模型，其中$ 3 $ -RICH线的数量与Green-Tao（OR，Burr，Grünbaum和Sloane）给出的预期数量一致，以最大的线路数。我们还显示，使用有限字段的椭圆曲线，存在无限的许多点线配置，其中$ 3 $ - 富含$ 3 $的线超过了绿色-TAO公式给出的预期数量，而这是唯一的最佳可能性，此外，唯一的最佳可能性与$ 3 $ rich Lines符合绿色TAO配方的$ 3 $。
Consider a set of $ n $ points on a plane. A line containing exactly $ 3 $ out of the $ n $ points is called a $ 3 $-rich line. The classical orchard problem asks for a configuration of the $ n $ points on the plane that maximizes the number of $ 3 $-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of $ 3 $-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of $ 3 $-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of $ 3 $-rich lines agrees with the Green-Tao formula.
