  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.
