Zulfiya Zhanuzakova

Differential equations encountered in various applications can be interpreted as equations in graphs. Using the simplest properties of graphs, it is possible to investigate the behavior of solutions to differential equations. Boundary value problems on graphs are defined by function spaces defined on graphs and differential systems. This article discusses the transformation of a boundary value problem in a graph into a boundary value problem for a differential system. The differential equation in the graphs is redefined by transforming each edge of the graph into the interval (0, 1). The boundary conditions are then also transformed according to the interval to establish a connection between the initial boundary value problem and the newly obtained boundary value problem.

The algebraic variety of Keller polynomials solves the Jacobian problem [1], [2], [3], [4], plays an important role in the solution. It is known that an algebraic variety of two Keller polynomials with two variables is a finite set [5]. This is confirmed by the calculation of the result of polynomials from two variables. Since the Keller polynomials in two variables are simple to each other, their results are different from zero. In addition, it uses Gauss’s value lemma over ring polynomials. Previously, a simple polynomial over a ring remained simple and over the field encompassing that ring. Now the coprime polynomials over the ring remain coprime and over the field enclosing this ring. Using this fact, we have shown in our work [5] that two Keller polynomials in two variables have a finite algebraic variety. In this article, we will show that for any variables, the algebraic polynomial value of Keller polynomials consists of only one point. The primary method is the analytical geometry coordinate method. It is understood that n variables of n Keller polynomials give an n -dimensional surface in the affine space 2n. The coordinates of any point on this surface are calculated on the n + 1-dimensional plane and on the n -dimensional surface. These coordinates are equated to each other. The crucial step of this proof – obtaining a derivative from polynomials in the n + 1 coordinate of this point – allows the use of the Jacobian matrix of the polynomial representation defined from given Keller polynomials.