Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+
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 ≥ 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 ℤ+. The principal aim of the present work is accordingly to develop a method for classifying families of algebraic curves having no solutions in 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 ∈ M3k × m(ℤ) satisfies suitable conditions on its entries, and let S={s1,s2,…,sN} ⊂ ℤm denote the solution set of the system, with A=(aij), and si=(c1,c2,…,cn+1). From such a solution we construct algebraic curves of degree n ≤ m, nonsingular curves Cjsi of genus g ≥ 0, of the form Cjsi:Y2=a3j1c1Xn+a3j2c2Xn−1+…+a3jncn. If Y ≥ 1, X > 1 and ∀(X,Y) ∈ Cjsi(ℤ+) holds then [Cjsi(ℤ+)]1 ≤ j ≤ kN=∅ and if the system S admits infinitely many solutions, then [Cjsi(ℤ+)]1 ≤ j ≤ k∞=∅ as N ⟶ ∞.
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BibTeX
@misc{fadul2026linear,
title = {{Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+}},
author = {Shazali Abdalla Fadul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27049},
eprint = {2609.27049},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27049},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+
AU - Shazali Abdalla Fadul
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27049
UR - https://arxiv.org/abs/2609.27049
AB - 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 ≥ 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 ℤ^+. The principal aim of the present work is accordingly to develop a method for classifying families of algebraic curves having no solutions in 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 ∈ M_3k × m(ℤ) satisfies suitable conditions on its entries, and let S={s_1,s_2,…,s_N} ⊂ ℤ^m denote the solution set of the system, with A=(a_ij), and s_i=(c_1,c_2,…,c_n+1). From such a solution we construct algebraic curves of degree n ≤ m, nonsingular curves C_js_i of genus g ≥ 0, of the form C_js_i:Y^2=a_3j1c_1X^n+a_3j2c_2X^n−1+…+a_3jnc_n. If Y ≥ 1, X > 1 and ∀(X,Y) ∈ C_js_i(ℤ^+) holds then [C_js_i(ℤ^+)]_1 ≤ j ≤ k^N=∅ and if the system S admits infinitely many solutions, then [C_js_i(ℤ^+)]_1 ≤ j ≤ k^∞=∅ as N ⟶ ∞.
ER -