University Proceedings. Volga Region. Physical and Mathematical Sciences

Abstract

Background. Differential connections between solutions of systems of differential equations play a significant role in mathematics and mathematical physics. The operators and algebras of differential symmetry of linear homogeneous systems of differential equations generated by such connections are of great importance. The conditions for the coincidence of internal and external algebras of differential symmetry lead to the concept of a theorem on the zeros of linear differential operators. The purpose of the work is to give a clear definition of the concept of a theorem on zeros for a family of possible, in particular, formal, solutions to a system of equations, to prove a general theorem on the division of linear differential operators for a family of formal solutions. Materials and methods. The necessary notations and concepts are introduced. A definition of the theorem on zeros of linear differential operators is given, and the analogy with Hilbert’s theorem is explained. The previously established conditions equivalent to the zero theorem and the connection with the conditions for the coincidence of external and internal differential symmetry algebras are discussed. When proving the formal theorem on the division of linear differential operators, elements of the theory of linear locally convex spaces are used. Results. The concept of a zero theorem is extended to the family of linear spaces of possible solutions to a system of differential equations, and global, local and formal zero theorems are defined. A theorem on conditions equivalent to the zero theorem is proved. The general concept of division of linear differential operators is introduced, a formal division theorem is formulated and proved, in which the coefficients of the resulting linear differential operator can be arbitrary functions. Conclusions. The results obtained in this work can serve as the basis for proving a number of theorems on the zeros of linear differential operators, in particular, formal theorems over the ring of infinitely differentiable functions.

Keywords

• linear differential operator
• coefficient ring
• homogeneous equation
• differential symmetry algebras
• zero theorems
• locally convex space