Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying 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 ≥ 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 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 } ⊂ Z m denote the solution set of the system, with si = (c1, c2, . . . , cn+1) and Dj ≥ 1 ∈ N, A = (aij ). From such a solution we construct algebraic curves of degree n ≤ m, of the form Cjsi : DjY 2 = a3j1c1Xn + a3j2c2Xn−1 − · · · + a3jn+1cn+1. If y ≥ 1, x > 1 and ∀(x, y) ∈ Cjs1 (Z +) holds then - Cjsi (Z +) N i=1 1≤j≤k = ∅ and if the system S admits infinitely many solutions, then - Cjsi (Z +) ∞ i=1 1≤j≤k = ∅ if N → ∞. We further extend this correspondence to several systems considered simultaneously. In particular, we consider three systems Sj : Ajx = bj , j = 1, 2, 3, representing linear, quadratic, and cubic conditions, respectively, with integer matrices. When these systems admit a common solution set S1 ∩ S2 ∩ S3 = {s1, s2, . . . , sN } ⊂ Z m and si = (x1, x2, . . . , xm), we construct an algebraic curve of degree deg Csi = Max(x1, x2, . . . , xm) and whose equation has the form Csi : DY 2 = a1Xx1 + a2Xx2 + · · · + amXxm. If x > 1, y ≥ 1 and ∀(x, y) ∈ Csi (Z +) holds then Csi (Z +) N i=1 = ∅. Several equivalent formulations of the conditions imposed on these systems are also given. The method is then generalized to common solutions of three algebraic curves defined by multivariable equations C1, C2, F3 ∈ Z(x1, . . . , xn). From a common solution of these three curves we construct a fourth algebraic curve in two variables CP ∈ Z(x, y), having no solution in Z +. Finally, we study intersections of algebraic curves. We prove that, given a solution of a linear equation of the form SM = {P1, P2, . . . , PN } ⊂ Z m+2 , one can construct three algebraic curves, denoted by CPi , GPi , FPi ∈ Z(x, y), whose common intersection over Z + is empty. Then CPi (Z +) ∩ GPi (Z +) ∩ FPi (Z +) ∞ i=1 = ∅ if N → ∞. The forms and degrees of the three curves so constructed are described explicitly.