Asymptotic estimates for roots of the cuboid characteristic equation in the nonlinear region

TL;DR

This paper investigates the asymptotic behavior of roots of a complex Diophantine equation related to perfect cuboids, introducing a nonlinear parameter combination to extend previous asymptotic analyses.

Contribution

It introduces a new nonlinear parameter combination for asymptotic estimates of roots of the cuboid characteristic equation, expanding on prior linear-based approaches.

Findings

Derived new asymptotic estimates for roots with nonlinear parameter combinations

Extended understanding of root behavior in the nonlinear parameter regime

Provided insights into the structure of the cuboid characteristic equation

Abstract

A perfect cuboid is a rectangular parallelepiped. Its edges, its face diagonals, and its space diagonal are of integer lengths. None of such cuboids is known thus far, though the system of Diophantine equations describing them is easily written. The cuboid characteristic equation is a twelfth degree Diophantine equation derived from the initial cuboid equations and equivalent to them. In the case of the second cuboid conjecture it reduces to a tenth degree equation. This equation comprises two parameters. Previously various asymptotics for roots of this equation were studied as its parameters tend to infinity either separately or simultaneously provided some linear combination of them is preserved finite. In the present paper this linear combination is replaced by a certain nonlinear expression.