Multidimensional versions of Poincaré's theorem for difference equations

A generalization to several variables of the classical Poincaré 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 Poincaré 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.
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