{
  "abstract": "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 $N \\longrightarrow \\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27049",
  "author_total": 1,
  "authors": [
    {
      "name": "Shazali Abdalla Fadul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b3zp7f4t5b5b0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27049",
  "fulltext_url": null,
  "id": "b3zp7f4t5b5b0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27049",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214186408",
  "pdf_url": "https://arxiv.org/pdf/2609.27049",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b3zp7f4t5b5b0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27049",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27049"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+",
  "title_text": "Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/b3zp7f4t5b5b0",
  "venue": "arXiv",
  "versions": []
}