A variant of the formal theorem on the zeros of linear differential operators

Background. In the theory of linear differential equations, transformations generated by differential substitutions of dependent variables play an essential role. The study of these transformations led to create a general theory of differential symmetry algebras of homogeneous linear systems of differential equations and to the theory of differential homomorphisms. These theories turned out to be closely related to the concept of the theorem on the zeros of linear differential operators (LDO). To date, several theorems on the zeros of linear equations have been proved, but these theorems are not sufficient to study the algebras of differential symmetry and the relations between different types of linear homogeneous systems of differential equations. The formulation and proof of new theorems about the zeros of LDO is an urgent task. The main objective of the work is to formulate and prove a version of the formal theorem on the zeros of linear differential operators. Another important objective is to construct examples of the theorem’s application that confirm its usefulness and validity. Materials and methods. Section 1 (Introduction) provides general information on the works that present theorems on zeros of LDO. The meaning of formal theorems on zeros and the role that such particular theorems can play in the general theory are explained. Section 2 presents the basic notations and concepts, and provides a definition of the theorem on zeros of linear differential operators for a family of modules over the ring of scalar linear differential operators. Section 3 describes the elements of the theory of pseudoinverses of matrices and operators, which are used in proving the main theorem of the work. Results. Section 4 formulates and proves a version of the formal theorem on zeros (Theorem 1). Section 5 gives examples of families of linear differential operators for which the conditions of Theorem 1 are satisfied (Theorems 2, 3, 4). In addition, a method for constructing local sections in the general pseudoinverse problem is described; a pseudoinverse matrix is applied in a new situation; a special basis is used in which the coordinates of the LDO coincide with its coefficients; a useful concept of the matrix of the main symbols of the LDO by columns is introduced. Conclusions. The results of the work can serve as the basis for proving the validity of the formal theorem on zeros for a set of specific linear differential operators and families of operators.