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 possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. 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.

本文的目的是研究有限域上投影线上的两个线性集之间的相交问题。特别是，我们分析了最大线性度可能不同的两个球杆之间的交点。我们还考虑了最大等级的某个线性集与相同等级的任何其他线性集之间的交集。该策略依赖于某些代数曲线的研究，这些代数曲线的有理点描述了两个线性集的交点。在其他几何和代数工具中，功能场理论和Hasse-Weil界起着至关重要的作用。作为应用，我们在BEL秩2的半场上给出渐近结果。