论文标题
在投影线中的线性集的交叉点问题上
On the intersection problem for linear sets in the projective line
论文作者
论文摘要
本文的目的是研究有限字段的投影线中的两个线性集之间的相交问题。特别是，我们分析了两个俱乐部之间的交集，最终具有不同的线性最大场。另外，我们分析了由多项式$αX^{q^k}+βx$定义的线性集之间的相交与具有相同等级的其他线性集合之间的相交；该家族包含由$ x^q $定义的伪古鲁斯类型的线性集。该策略依赖于某些代数曲线的研究，其理性点描述了两个线性集的相交。在其他几何和代数工具中，功能场理论和HASSE-WEIL BOUND起着至关重要的作用。作为一种应用，我们在Bel-Rask二的半场上给出渐近结果。
The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with eventually different maximum fields of linearity. Also, we analyze the intersection between the linear set defined by the polynomial $αx^{q^k}+βx$ and other linear sets having the same rank; this family contains the linear set of pseudoregulus type defined by $x^q$. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse-Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.
