Solvability of a difference Cauchy problem for multi-layer implicit difference schemes

Difference equations arise in different areas of mathematics. Difference equations in conjunction with a method of generation functions give a efficient technique for studying the enumerative problems in the combinatorial analyses. Another source of difference equations is discretization of differential equations. Methods of discretization a differential equation is an important part of the theory of difference schemes, and also lead to difference equations [1]. In the case of implicit difference schemes its solvability presents a non-trivial question. In [2] investigated the stability of a two-layer homogeneous linear difference scheme with constant coefficients. In [3] to study the stability of multilayer homogeneous difference schemes applied theory of amoebas of algebraic hypersurfaces and a formula for the solution of the Cauchy problem in terms of its fundamental solution. In [4] for the two-dimensional case is investigated difference analog of the boundary value problem for Hormander polynomial differential operator. We investigate the solvability of difference equations with initial-boundary conditions of Riquier and consider them as implicit multi-layer difference schemes. Since this question reduces to solvability of systems of linear equations, we use linear algebra to give necessary and sufficient conditions and a simple sufficient condition for solvability in terms of coefficients of a polynomial difference operator. We show the relation of these results to the elimination algorithm for systems of algebraic equations with band matrices. The results can be applied for studying solvability of difference schemes and construction of monomial bases in quotients of the polynomial ring.