Estimating of the number of natural solutions of homogeneous algebraic Diophantine diagonal equations with integer coefficients

TL;DR

The paper introduces a new method for estimating the number of natural solutions to homogeneous algebraic Diophantine equations, providing both lower and upper bounds, and compares these estimates to existing methods like the circle method.

Contribution

A novel approach for lower estimation of solutions to Diophantine equations, extending bounds for equations with any number of variables and analyzing the relation between upper and lower estimates.

Findings

Developed a new method for lower bounds of solutions.

Established upper bounds using the circle method for certain degrees.

Analyzed the relationship between upper and lower estimates.

Abstract

Author developed a method in the paper, which, unlike the circle method of Hardy and Littlewood (CM), allows you to perform a lower estimate for the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. It was found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coefficients diagonal form with any number of variables using this method. Author obtained upper bound of the number of the natural solutions (using CM) of one type of homogeneous Diophantine equation for values $k \geq \log_2 s$, where $k$ is the degree of the equation and $s$ is the number of variables. It was also found the upper bound of the number of the natural solutions of the homogeneous algebraic Diophantine equation with integer coefficients with a small number of variables. Author investigated the relations of upper and lower estimates of the number of natural solutions of homogeneous Diophantine equation with integer coefficients diagonal form in the paper.