Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+
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 $N \longrightarrow \infty$.