Estimations of the number of solutions of algebraic Diophantine equations with natural coefficients using the circle method of Hardy-Littlewood

Abstract: This article discusses the question - how to estimate the number of solutions of algebraic Diophantine equations with natural coefficients using Circular method developed by Hardy and Littlewood. This paper considers the estimate of the number of solutions of algebraic Diophantine equation: $c_1x_1^{k_1}+c_2x_2^{k_2}+...+c_sx_s^{k_s}=n$. The author found the asymptotic estimate for the number of solutions of this equation as a function of the value $n$, if all coefficients and $n$ are natural. This article analyzes the results and shows that these estimates of the number of natural solutions of the equations have high accuracy.