Abstract:
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.
Citation:
E. K. Leinartas, M. Passare, A. K. Tsikh, “Multidimensional versions of Poincaré's theorem for difference equations”, Sb. Math., 199:10 (2008), 1505–1521
This publication is cited in the following 14 articles:
Svetlana S. Akhtamova, Tom Cuchta, Alexander P. Lyapin, “An Approach to Multidimensional Discrete Generating Series”, Mathematics, 12:1 (2024), 143
Evgeny D. Leinartas, August K. Tsikh, “On a multidimensional version of the principal theorem of difference equations with constant coefficients”, Zhurn. SFU. Ser. Matem. i fiz., 15:1 (2022), 125–132
A. P. Lyapin, S. S. Akhtamova, “Rekurrentnye sootnosheniya dlya sechenii proizvodyaschego ryada resheniya mnogomernogo raznostnogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 31:3 (2021), 414–423
Kytmanov A.A. Lyapin A.P. Sadykov T.M., “Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients”, Program. Comput. Softw., 43:2 (2017), 105–111
E. K. Leǐnartas, M. S. Rogozina, “Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring”, Siberian Math. J., 56:1 (2015), 92–100
Mikhalkin E.N., Shchuplev A.V., Tsikh A.K., “Amoebas of Cuspidal Strata for Classical Discriminant”, Complex Analysis and Geometry, Springer Proceedings in Mathematics & Statistics, Springer Proceedings in Mathematics & Statistics, 144, eds. Bracci F., Byun J., Gaussier H., Hirachi K., Kim K., Shcherbina N., Springer, 2015, 257–272
Natalia A. Bushueva, Konstantin V. Kuzvesov, Avgust K. Tsikh, “On the asymptotic of homological solutions to linear multidimensional difference equations”, Zhurn. SFU. Ser. Matem. i fiz., 7:4 (2014), 417–430
Passare M., Pochekutov D., Tsikh A., “Amoebas of Complex Hypersurfaces in Statistical Thermodynamics”, Math. Phys. Anal. Geom., 16:1 (2013), 89–108
Marina S. Rogozina, “Ustoichivost mnogosloinykh raznostnykh skhem i ameby algebraicheskikh giperpoverkhnostei”, Zhurn. SFU. Ser. Matem. i fiz., 5:2 (2012), 256–263
N. A. Bushueva, A. K. Tsikh, “On amoebas of algebraic sets of higher codimension”, Proc. Steklov Inst. Math., 279 (2012), 52–63
E. K. Leǐnartas, “Stability of the Cauchy problem for a multidimensional difference operator and the amoeba of the characteristic set”, Siberian Math. J., 52:5 (2011), 864–870
Aleksandr P. Lyapin, “Posledovatelnosti Riordana i dvumernye raznostnye uravneniya”, Zhurn. SFU. Ser. Matem. i fiz., 2:2 (2009), 210–220
Evgenii K. Leinartas, Aleksandr P. Lyapin, “O ratsionalnosti mnogomernykh vozvratnykh stepennykh ryadov”, Zhurn. SFU. Ser. Matem. i fiz., 2:4 (2009), 449–455
D. Yu. Pochekutov, “Diagonals of the Laurent series of rational functions”, Siberian Math. J., 50:6 (2009), 1081–1091
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