Stability of multilayer inhomogeneous difference schemes and amoebas of algebraic hypersurfaces

We study the stability of polynomial difference operators coming mainly from the theory of difference schemes. In the study we use the terminology and the methods of this theory as well as those of the theory of amoebas of algebraic hypersurfaces. The notion of amoeba allows to formulate a multidimensional analog of the condition that all roots of the characteristic polynomial lie in the unit disc, i.e. the stability condition for multidimensional difference schemes. In the terms of the latter we formulate and prove the stability criterion for multilayer linear inhomogeneous difference schemes. A formula representing the solution to the Cauchy problem via its fundamental solution is obtained.