Integer conversions and estimation of the number of integer solutions of algebraic Diophantine equations

TL;DR

This paper investigates the maximum number of integer solutions for certain algebraic Diophantine equations, introducing integer conversions that preserve solution asymptotics and estimating solutions for nondiagonal forms.

Contribution

It introduces integer conversions that maintain asymptotic solution counts and provides estimates for solutions of nondiagonal algebraic Diophantine equations.

Findings

Integer conversions preserve asymptotic behavior of solutions.

Estimated the number of solutions for nondiagonal equations.

Analyzed equations with coefficients of opposite signs.

Abstract

The paper assesses the top number of integer solutions for algebraic Diophantine Thue diagonal equation of the degree $n \geq 2$ and number of variables $k > 2$ and equations with explicit variable in the case when the coefficients of the equation are of the opposite signs. The author found integer conversions that maintain the asymptotic behavior of the number of integer solutions of algebraic Diophantine equation in the case of the conversion equation to diagonal form. The paper considers the estimation of the number of integer solutions for some types of algebraic Diophantine equations with nondiagonal form.