Applied Physics and Mathematics Annotation << Back ESTIMATING OF THE NUMBER OF INTEGER (NATURAL) SOLUTIONS OF INHOMOGENEOUS ALGEBRAIC DIOPHANTINE DIAGONAL EQUATIONS WITH INTEGER COEFFICIENTS V.L. VOLFSON This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coeffi cients without a free member via the circle method of Hardy and Littlewood (KM). Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. It was developed a method in the paper, which allows you to perform the low estimate of the number of integer (natural) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estimate (with this method) of the number of integer (natural) solutions for certain kinds of inhomogeneous algebraic Diophantine diagonal equations with integer coeffi cients with any number of variables (including Thue’s equation). Keywords: inhomogeneous algebraic Diophantine equation, Thue’s equation, integer coefficients, constant term, diagonal, circle method of Hardy and Littlewood, Hua’s lemma, number of solutions, natural solutions, integer solutions, upper bound, lower bound, hypercube. Contacts: E-mail: znakvicvolf@mail.ru Pp. 21-28.