Multidimensional versions of Poincare's theorem for difference equations

A generalization to several variables of the classical Poincare theorem on the asymptotic behaviour of solutions of a linear difference equation is presented. Two versions are considered: 1) general solutions of a system of n equations with respect to a function of n variables and 2) special solutions of a scalar equation. The classical Poincare theorem presumes that all the zeros of the. limiting symbol have different. absolute values. Using the notion of an amoeba of an algebraic hypersurface, a multidimensional analogue of this property is formulated; it ensures nice asymptotic behaviour of special solutions of the corresponding difference equation.