Applied Physics and Mathematics Annotation << Back ESTIMATING OF THE NUMBER OF THE NATURAL SOLUTIONS OF HOMOGENEOUS ALGEBRAIC DIOPHANTINE EQUATIONS OF THE DIAGONAL FORM WITH INTEGER COEFFICIENTS V.L. VOLFSON It was 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 coeffi cients. Author found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coeffi cients diagonal form with any number of variables using this method. It was obtained upper bound of the number of the natural solutions of one type of homogeneous Diophantine equation for values k ≥ 1 + log2s, where k is the degree of the equation, s is the number of variables using CM. Author also found the upper bound of the number of the natural solutions of the homogeneous algebraic Diophantine equation with integer coeffi cients with a small number of variables. It was investigated of the relations of upper and lower estimates of the number of natural solutions of homogeneous Diophantine equation with integer coeffi cients diagonal form in the paper. Keywords: homogeneous algebraic Diophantine equation, integer coeffi cients, diagonal, circle method of Hardy and Littlewood, lemma Hua, the number of solutions, organic solutions, upper estimate, lower estimate, asymptotic estimate. Contacts: E-mail: znakvicvolf@mail.ru Pp. 15-26.