Geodesic
Multidimensional versions of Poincar\'e's theorem for difference equations
Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1505-1521

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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. Bibliography: 20 titles. 
@article{SM_2008_199_10_a4,
     author = {E. K. Leinartas and M. Passare and A. K. Tsikh},
     title = {Multidimensional versions of {Poincar\'e's} theorem for difference equations},
     journal = {Sbornik. Mathematics},
     pages = {1505--1521},
     publisher = {mathdoc},
     volume = {199},
     number = {10},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_10_a4/}
}
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E. K. Leinartas; M. Passare; A. K. Tsikh. Multidimensional versions of Poincar\'e's theorem for difference equations. Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1505-1521. http://geodesic.mathdoc.fr/item/SM_2008_199_10_a4/