Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring

The Cauchy Problem for Multidimensional Difference Equations and the Preservation of the Hierarchy of Generating Functions of its Solutions

“…has a unique solution in an obvious way. For n > 1, the situation is much more complicated and the difficulties are related to the fact that the solution space of the difference equation is infinite-dimensional, and the question of additional conditions that allow to single out unique solutions in this space is non-trivial (see, for example, [1,2,5,8]).…”

“…Conditions on the cone and the point m that ensure solvability, that is, existence and uniqueness, have been studied in papers [2,5,8,13].…”

On the Correctness of Polynomial Difference Operators

“…It is true (see [11]) easily verifiable sufficient condition for solvability of the problem (5)- (6).…”

Correctness of a Two-dimensional Cauchy Problem for a Polynomial Difference Operator with Constant Coefficients

“…A difference analog of the boundary value problem of Hörmander for a polynomial differential operator has been studied in [12] in two-dimensional case and in [13] for arbitrary number of variables. In [13] we have studied solvability of difference equations with initial-boundary Riquiertype conditions; in terms of the theory of difference schemes they are multilayer implicit difference schemes.…”

“…In [13] we have studied solvability of difference equations with initial-boundary Riquiertype conditions; in terms of the theory of difference schemes they are multilayer implicit difference schemes. In [14] an easily verifiable sufficient condition for correctness of a Cauchy problem for a polynomial difference operator with constant coefficients whose characteristic polynomial is homogeneous has been obtained.…”