Transverse-freeness in finite geometries

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-06-03 DOI:10.1016/j.ffa.2025.102663

Charlie Bruggemann , Vera Choi , Brian Freidin , Jaedon Whyte

引用次数: 0

Abstract

We study the interplay between combinatorial and algebraic geometry via projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending the 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in n-dimensional projective space that are tangent to every k-dimensional subspace, for some values of n and k. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces, we prove existence results for low-degree curves tangent to each of the lowest degree curves defined over the base field.

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有限几何中的横向自由

我们研究了组合几何和代数几何之间的相互作用，通过定义在有限域上的射影曲线和超曲面，它们与一类低次变的每一个成员相切。扩展Asgarli的二维工作，我们首先探索了n维射影空间中与每个k维子空间相切的光滑超曲面所能达到的最低度，对于某些n和k值。然后我们研究了作为有限逆平面和双曲平面模型的射影曲面，以及球面和双曲几何的有限类似物。在这些曲面上，我们证明了与基域上定义的每条最低次曲线相切的低次曲线的存在性结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.