The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for algebraic Diophantine equation. The author shows the properties of this transformation and he receives the algorithm for finding the matrix elements of a generalized integer orthogonal transformation for algebraic Diophantine equation of the second order to diagonal form. The article includes examples illustrating the reduction of algebraic equations of the second order to the diagonal form with the help of integer generalized orthogonal transformation and of determination asymptotics behavior of integer solutions for these equations.
Integer conversions and estimation of the number of integer solutions of algebraic Diophantine equations
Estimations of the number of solutions of algebraic Diophantine equations with natural coefficients using the circle method of Hardy-Littlewood
On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n
![QR code for arXiv:1704.02238 [math.NT]](http://oss.arxitics.com/images/favicon.svg)