Converting of algebraic Diophantine equations to a diagonal form with the help of generalized integer orthogonal transformation, maintaining the asymptotic behavior of the number of its integer solutions

TL;DR

This paper introduces a generalized integer orthogonal transformation that converts second-order algebraic Diophantine equations into diagonal form while preserving the asymptotic count of solutions, aiding analysis and solution enumeration.

Contribution

It develops a new generalized integer orthogonal transformation method and provides an algorithm for diagonalizing second-order Diophantine equations.

Findings

Transformation preserves asymptotic behavior of solutions.

Algorithm effectively reduces equations to diagonal form.

Examples demonstrate practical application of the method.

Abstract

The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for algebraic Diophantine equation. The author shows the properties of this transformation and he receives the algorithm for finding the matrix elements of a generalized integer orthogonal transformation for algebraic Diophantine equation of the second order to diagonal form. The article includes examples illustrating the reduction of algebraic equations of the second order to the diagonal form with the help of integer generalized orthogonal transformation and of determination asymptotics behavior of integer solutions for these equations.