线性代数与代数几何：一种用于分类在 Z^+ 中无解的代数曲线族的矩阵构造
Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+
- Faculty Of Mathematics Sciences & Statistics, AL-neelain University（阿尔尼林大学数学科学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过建立代数几何与线性代数之间的对应，利用线性系统构造非奇异代数曲线，提出一种系统分类在正整数集上无解的代数曲线族的方法。
AI中文摘要：
关于次数 d=3 的曲线（即椭圆曲线）已有大量已知结果，包括 Siegel、Mazur 和 Mordell 等人的定理。更一般地，Faltings 对所有次数 d≥2 的代数曲线建立了全面的结果。然而，这些结果通常并不提供一种显式或系统性的程序来分类在给定集合（如 Z^+）上无解的代数曲线。因此，本工作的主要目标是发展一种方法来分类在 Z^+ 中无解的代数曲线族。我们首先建立代数几何与线性代数之间的对应关系，并由此发展一种基于线性代数构造和工具的分类程序。具体而言，设 S:Ax=b 为一个线性系统，其中 A∈M_{3k×m}(Z) 满足其条目上的适当条件，并设 S={s₁,s₂,…,s_N}⊂Z^m 为该系统的解集，其中 A=(a_{ij})，s_i=(c₁,c₂,…,c_{n+1})。从这样的解出发，我们构造次数 n≤m 的代数曲线，即亏格 g≥0 的非奇异曲线 C_{js_i}，形式为 C_{js_i}:Y²=a_{3j1}c₁X^n+a_{3j2}c₂X^{n-1}+…+a_{3jn}c_n。若 Y≥1，X>1 且对 C_{js_i}(Z^+) 中的所有 (X,Y) 成立，则 [C_{js_i}(Z^+)]_{1≤j≤k}^N=∅，并且若系统 S 有无穷多解，则当 N→∞ 时，[C_{js_i}(Z^+)]_{1≤j≤k}^∞=∅。
英文摘要：
A substantial body of results is known for curves of degree $d=3$, namely elliptic curves, including the theorems of Siegel, Mazur, and Mordell, among others. More generally, Faltings established comprehensive results for all algebraic curves of degree $d \ge 2$. However, these results do not, in general, furnish an explicit or systematic procedure for classifying algebraic curves that fail to admit solutions over a prescribed set such as $\mathbb{Z}^{+}$. The principal aim of the present work is accordingly to develop a method for classifying families of algebraic curves having no solutions in $\mathbb{Z}^{+}$. We begin by establishing a correspondence between algebraic geometry and linear algebra, from which we develop a classification procedure grounded in linear-algebraic constructions and tools. Specifically, let $S:Ax=b$ be a linear system, where $A \in M_{3k \times m}(\mathbb{Z})$ satisfies suitable conditions on its entries, and let $S=\{s_{1},s_{2},\dots,s_{N}\} \subset \mathbb{Z}^{m}$ denote the solution set of the system, with $A=(a_{ij})$, and $s_{i}=(c_{1},c_{2},\dots,c_{n+1})$. From such a solution we construct algebraic curves of degree $n \le m$, nonsingular curves $C_{js_i}$ of genus $g \ge 0$, of the form $C_{js_{i}}:Y^{2}=a_{3j1}c_{1}X^{n}+a_{3j2}c_{2}X^{n-1}+\dots+a_{3jn}c_{n}$. If $Y \ge 1, X > 1$ and $\forall(X,Y) \in C_{js_{i}}(\mathbb{Z}^{+})$ holds then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{N}=\emptyset$ and if the system $S$ admits infinitely many solutions, then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{\infty}=\emptyset$ as $N \longrightarrow \infty$.
