Differential transcendence of solutions of systems of linear differential equations based on total reduction of the system

Ivana Jovovic

doi:10.2298/aadm190627024j

Abstract

In this paper we consider total reduction of the nonhomogeneous linear system of operator equations with constant coefficients and commuting operators. The totally reduced system obtained in this manner is completely decoupled. All equations of the system differ only in the variables and in the nonhomogeneous terms. The homogeneous parts are obtained using the generalized characteristic polynomial of the system matrix. We also indicate how this technique may be used to examine differential transcendence of the solution of the linear system of the differential equations with constant coefficients over the complex field and meromorphic free terms.

Highlights

One way to solve a linear system of operator equations with constant coefficients is to decompose it into several subsystems using Jordan canonical form
Every subsystem of the reduced system corresponds to one block of Jordan canonical form of the system matrix
The rational canonical form of a matrix is the best diagonal block form that can be obtained over the field of coefficients and it corresponds to the factorization of the characteristic polynomial into invariant factors without adding any field extension

Summary

INTRODUCTION

One way to solve a linear system of operator equations with constant coefficients is to decompose it into several subsystems using Jordan canonical form. Total reduction of the linear systems of the operator equations, characteristic polynomial, generalized characteristic polynomial, sum of principal minors, differential transcendence. Another method for solving a linear system of first-order operator equations which does not require a change of basis is discussed in [10]. The system is reduced to a totally reduced system, i.e., to a system with separated variables, by using the characteristic polynomial ∆B(λ) = det(λI − B) of the system matrix B This system consists of higher-order operator equations which differ in variables and nonhomogeneous terms.

The generalized characteristic polynomial

The main result

Transcendental and differentially algebraic extensions