Estimating of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients

TL;DR

This paper develops a method using the circle technique to estimate upper and lower bounds on the number of integer solutions to inhomogeneous algebraic Diophantine diagonal equations with integer coefficients, including Thue equations.

Contribution

The paper introduces a new method for estimating the number of solutions, providing explicit bounds for inhomogeneous algebraic Diophantine diagonal equations with any number of variables.

Findings

Established an upper bound for solutions using the circle method.

Derived a lower estimate for solutions of certain inhomogeneous equations.

Applicable to equations with any number of variables, including Thue equations.

Abstract

This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estimate (with this method) of the number of integer (natural) solutions for certain kinds of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients with any number of variables (including Thue equation).