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Mat. Sb., 2008, Volume 199, Number 10, Pages 87–104 (Mi msb4503)  

This article is cited in 11 scientific papers (total in 11 papers)

Multidimensional versions of Poincaré's theorem for difference equations

E. K. Leinartasa, M. Passareb, A. K. Tsikha

a Siberian Federal University
b Stockholm University

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.

DOI: https://doi.org/10.4213/sm4503

Full text: PDF file (621 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2008, 199:10, 1505–1521

Bibliographic databases:

UDC: 517.55+517.965
MSC: Primary 39A11; Secondary 32A60
Received: 27.12.2007

Citation: E. K. Leinartas, M. Passare, A. K. Tsikh, “Multidimensional versions of Poincaré's theorem for difference equations”, Mat. Sb., 199:10 (2008), 87–104; Sb. Math., 199:10 (2008), 1505–1521

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Aleksandr P. Lyapin, “Posledovatelnosti Riordana i dvumernye raznostnye uravneniya”, Zhurn. SFU. Ser. Matem. i fiz., 2:2 (2009), 210–220  mathnet  elib
    2. Evgenii K. Leinartas, Aleksandr P. Lyapin, “O ratsionalnosti mnogomernykh vozvratnykh stepennykh ryadov”, Zhurn. SFU. Ser. Matem. i fiz., 2:4 (2009), 449–455  mathnet  elib
    3. D. Yu. Pochekutov, “Diagonals of the Laurent series of rational functions”, Siberian Math. J., 50:6 (2009), 1081–1091  mathnet  crossref  mathscinet  isi  elib  elib
    4. 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  mathnet  crossref  mathscinet  isi
    5. Marina S. Rogozina, “Ustoichivost mnogosloinykh raznostnykh skhem i ameby algebraicheskikh giperpoverkhnostei”, Zhurn. SFU. Ser. Matem. i fiz., 5:2 (2012), 256–263  mathnet
    6. N. A. Bushueva, A. K. Tsikh, “On amoebas of algebraic sets of higher codimension”, Proc. Steklov Inst. Math., 279 (2012), 52–63  mathnet  crossref  mathscinet  isi  elib
    7. Passare M., Pochekutov D., Tsikh A., “Amoebas of Complex Hypersurfaces in Statistical Thermodynamics”, Math. Phys. Anal. Geom., 16:1 (2013), 89–108  crossref  mathscinet  zmath  isi  elib  scopus
    8. 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  mathnet
    9. 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  mathnet  crossref  mathscinet  isi  elib  elib
    10. 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  crossref  mathscinet  zmath  isi  scopus
    11. 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  crossref  mathscinet  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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