Quantitative indicators of the solutions of Diophantine equations and   systems in the domain of the natural numbers
  The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $<x_1,...x_k>$ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometrically proved in the paper. The author proves the assertion about the number of solutions of algebraic Diophantine equations of higher orders in the domain of the natural numbers. The author provides the estimates for the asymptotic behavior of quantitative solutions of Diophantine equations and systems in the domain of the natural numbers. 
Victor Volfson
math.NT