The stability of a linear autonomous difference equation with two complex coefficients and different delays is investigated. The starting point of the study is the Schur-Cohn theorem on the location of the roots of the characteristic equation in the complex plane with respect to the unit disk. To construct the domain of the exponential stability of the considered equation in the parameter space, the method of D-decomposition is used, which consists in constructing surfaces in the phase space such that when these surfaces are crossed by a point moving in the phase space, then the number of roots of the corresponding to the point characteristic equation, located outside the unit disk in the complex plane, changes. The region to which the zero number of roots corresponds is the domain of stability. This scheme is implemented for the above-mentioned difference equation: geometric stability criteria are found and the domains of exponential stability in the four-dimensional space of coefficients are described. Uniform stability is studied separately. Its domain is the domain of exponential stability supplemented by a part of its boundary. For the exact description of the domain of uniform stability, the description of a “multiplicity curve” was required, that is the line, all points of which correspond to multiple roots of the characteristic equation. The obtained results can be applied to the study of processes in physics, engineering, economics, and biology, for the modeling of which discrete models in the form of difference equations are used.

difference equations, exponential stability, uniform stability, D-decomposition.