论文标题
平面曲线引起有限场上的阻塞集
Plane curves giving rise to blocking sets over finite fields
论文作者
论文摘要
近年来，多项式方法的许多有用应用已在有限的几何形状中出现。实际上，代数曲线，尤其是由Rédei-Type多项式定义的曲线，在研究阻止集方面具有强大的功能。在本文中，我们逆转了发动机和研究，当阻断集合时可能是由有限场上平面曲线上的理性点产生的。我们表明，低度的不可还原曲线无法提供阻塞集，并证明了立方和四分之一曲线的更精致的结果。另一方面，使用数字理论的工具，我们构造了在$ \ mathbb {f} _p $ a guger of $ 4p^{3/4}+1 $中的平滑平面曲线，其点形成了阻塞集。
In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over $\mathbb{F}_p$ of degree at most $4p^{3/4}+1$ whose points form blocking sets.
