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

TL;DR

This paper extends methods to convert various algebraic second-order and higher-order Diophantine equations into diagonal form using integer non-orthogonal transformations, preserving the asymptotic count of solutions and providing estimates for solutions of diagonal Thue equations.

Contribution

It generalizes the diagonalization technique to a broader class of algebraic Diophantine equations and derives asymptotic estimates for their integer solutions.

Findings

Converted wider class of second-order equations to diagonal form

Provided asymptotic estimates for solutions of diagonal Thue equations

Extended transformation conditions to higher-order equations

Abstract

The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd degree with an amount of variables greater than two, and also he got and asymptotic estimates of the number of integer solutions of other types of diagonal algebraic Diophantine equations.