Applied Physics and Mathematics Annotation << Back ESTIMATIONS OF THE NUMBER OF SOLUTIONS OF ALGEBRAIC DIOPHANTINE EQUATIONS WITH NATURAL COEFFICIENTS USING THE CIRCLE METHOD OF HARDY-LITTLEWOOD V.L. VOLFSON It can be successfully used methods of complex function for estimations of the number of solutions of Diophantine equations. This article discusses the question – how to estimate the number of solutions of algebraic Diophantine equations with natural coeffi cients using the circular method developed by Hardy and Littlewood. This paper considers the estimate of the number of solutions of algebraic Diophantine equation: . The author found the asymptotic estimate for the number of solutions of this equation as a function of the value n, if all coeffi cients and n are natural. The article analyzes the results and shows that these estimates of the number of natural solutions of the equations have high accuracy. Keywords: algebraic Diophantine equation, Circle method of Hardy and Littlewood, Lemma Hua, asymptotic estimate, number of solutions, natural solutions. Contacts: E-mail: znakvicvolf@mail.ru Pp. 19-23.