Applied Physics and Mathematics Annotation << Back CONVERTING OF ALGEBRAIC DIOPHANTINE EQUATIONS TO A DIAGONAL FORM WITH THE HELP OF AN INTEGER NON-ORTHOGONAL TRANSFORMATION, MAINTAINING THE ASYMPTOTIC BEHAVIOR OF THE NUMBER OF ITS INTEGER SOLUTIONS V.L. VOLFSON Diophantine equations are used to study complex discrete systems, when looking for their optimal structures. The determination of the number of integer solutions of Diophantine equations is used in economics, for example, when determining the number of options for drawing up a plan, etc. The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd degree with an amount of variables greater than two, and also he got and asymptotic estimates of the number of integer solutions of other types of diagonal algebraic Diophantine equations. The article includes explanatory examples of transforming algebraic Diophantine equations to diagonal form using an integer non-orthogonal transformation and determining the asymptotic behavior of integer solutions of these equations. Key words: algebraic Diophantine equation, diagonal form, number of integer solutions, asymptotic estimate of the number of integer solutions, integer non-orthogonal transformation, transformation matrix. Contacts: E-mail: znakvicvolf@mail.ru Pp. 27-36.