Nondegenerate Canonical Solutions of one System of Functional Equations

Abstract

In this paper, we solve a special system of functional equations arising in the problem of embedding an additive two-metric phenomenologically symmetric geometry of two sets of rank (2,2) into a multiplicative two-metric phenomenologically symmetric geometry of two sets of rank (3,2). We are looking for non-degenerate solutions of this system, which are very difficult to determine in general terms. However, the problem of determining the set of its fundamental solutions associated with a finite number of Jordan forms of nonzero second-order matrices turned out to be much simpler and more meaningful in the mathematical sense. The methods developed by the authors can be applied to other systems of functional equations, the nondegenerate solutions of which prove the possibility of mutual embedding of some geometries of two sets.