Solution of a system of functional equations associated with an affine group

R. A. Bogdanova ; V. A. Kyrov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 3, pp. 24-32

Résumé

Solution of the embedding problem for a two-metric phenomenologically symmetric geometry of rank $(3,2)$ with the function $ g (x, y, \xi, \eta) = (g^{1}, g^{2 })= (x\xi+y\ mu,x\eta + y\nu)$ into an affine two-metric phenomenologically symmetric geometry of rank $(4,2)$ with the function $f(x,y,\xi,\eta,\mu,\nu)=(f^{1},f^{2})=(x\xi+y\mu+\rho,x\eta + y\nu+\tau)$ leads to the problem of establishing the existence of non-degenerate solutions to the corresponding system $f(\bar{x},\bar {y},\bar{\xi},\bar{\eta},\bar{\mu},\bar{\nu})=\chi(g(x,y,\xi,\eta),\mu,\nu)$ of two functional equations. This system is solved based on the fact that the functions $g$ and $f$ are previously known. This system is written explicitly as follows: $\bar{x}\bar{\xi }+\bar{y}\bar{\mu}+\bar{\rho}= \chi^{1}(x\xi +y\mu,x\eta+y\nu,\mu,\nu),$ $\bar{x}\bar{\eta }+\bar{y}\bar{\nu }+\bar{\tau}= \chi ^{2}(x\xi+y\mu,x\eta + y\nu,\mu,\nu).$ The main goal of this work is to find a general non-degenerate solution to this system. To solve the problem, we first differentiate with respect to the variables $x$, $y$ and $\xi$, $\eta$, $\mu$, $\nu$, as a result we obtain a system of differential equations with a matrix of coefficients $A$ of the general form. It is proved that the matrix $A$ can be reduced to Jordan form. Then a system of differential equations with such a Jordan matrix is solved. Returning to the original original system of functional equations, we find the additional restrictions. As a result, we arrive at a non-degenerate solution to the original system of functional equations.