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Implicit Linear First Order Difference Equations Over Commutative Rings

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

In this paper we study the problem of existence and uniqueness of solutions of implicit linear difference equations over commutative rings. The cases where the ring is integrally closed, Noetherian, local are considered. The question of periodic and quasi-polynomial solutions of this equation is also studied. In the case of a local ring, we obtain a formula for the solution of the equation as a series converging in \(\mathcal {M}\)-adic topology, where \(\mathcal {M}\) is the maximal ideal of the ring in question.

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References

  1. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)

    MATH  Google Scholar 

  2. Kelley, W.G.: Peterson, A.C., Difference Equation: An Introduction with Applications, 2nd ed., pp. 404. Academic Press (2001)

    Google Scholar 

  3. Halanay, A., Wexler, D.: Teoria Calitativa A Sistemelor Cu Impulsuri. Academiei Republicii Socialiste Romania, Bucuresti (1968)

    MATH  Google Scholar 

  4. Campbell, S.L.: Singular Systems of Differential Equations I, vol. 40. Pitman Publishing, Research Notes in Mathematics, San Francisko, London, Melbourne (1980)

    Google Scholar 

  5. Healton, J.W.: Discrete time systems, operator models and scattering theory. J. Funct. Anal. 16(1), 15–38 (1974)

    Article  MathSciNet  Google Scholar 

  6. Benabdallakh, M., Rutkas, A.G., Solov’ev, A.A.: Application of Asymptotic Expansions to the Investigation of an Infinite System of Equations, \(Ax_{n+1} + Bx_n = f_n\) in a Banach Space. J. Soviet Math. 48 (1990). Iss. 2, 124–130. https://doi.org/10.1007/BF01095789

  7. Bondarenko, M., Rutkas, A.: On a Class of Implicit Difference Equations. Dopov. Nac. Acad. Nauk Ukr., No. 7, 11–15 (1998)

    Google Scholar 

  8. Baskakov, A.G.: On the invertibility of linear difference operators with constant coefficients. Russ. Math. 45(5), 1–9 (2001)

    MathSciNet  Google Scholar 

  9. Gefter, S.L., Piven, A.L.: Implicit linear nonhomogeneous difference equation in banach and locally convex spaces. J. Math. Phys. Anal. Geom. 15, No. 3, 336–353 (2019)

    Google Scholar 

  10. Gerasimov, V.A., Gefter, S.L., Goncharuk, A.B.: Application of the \(p\)-Adic topology on \(\textbf{Z} \) to the problem of finding solutions in integers of an implicit linear difference equation. J. Math. Sci. 235, 256–261 (2018). https://doi.org/10.1007/s10958-018-4072-x

    Article  MathSciNet  MATH  Google Scholar 

  11. Gefter, S., Goncharuk, A.: Generalized backward shift operators on the ring \(\textbf{Z} [[x]]\), Cramer’s rule for infinite linear systems, and \(p\)-adic integers. Oper. Theory: Adv. Appl. 268, 247–259 (2018). https://doi.org/10.1007/978-3-319-75996-8_13

    Article  MathSciNet  Google Scholar 

  12. Martseniuk, V., Gefter, S.L., Piven’, A.: Uniqueness criterion and cramer’s rule for implicit higher order linear difference equations over \(\textbf{Z}\). In: Baigent, S., Bohner, M., Elaydi, S. (eds.), Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol. 341. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60107-2_16

  13. Matsumura, H.: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 2nd ed. Cambridge University Press (2006). https://doi.org/10.1017/CBO9781139171762

  14. Bourbaki, N.: Commutative Algebra, Hermann (1972)

    Google Scholar 

  15. Anderson, D.D., Anderson, D.F., Zaprullar, M.: Compeletely integrally closed Prufer \(v\)-multiplication Domains. Commun. Algebra 45(12), 5264–5282 (2017). https://doi.org/10.1080/00927872.2017.1303502

    Article  MATH  Google Scholar 

  16. Lang, S.: Algebra, Reviews, 3rd edn. Springer, New York (2002)

    Book  Google Scholar 

  17. McDonald, B.R.: Linear Algebra over Commutative Rings, 1st ed. CRC Press (1984). https://doi.org/10.1201/9781003065197

  18. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, New York (1990)

    Book  MATH  Google Scholar 

  19. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press (1996). https://doi.org/10.1017/CBO9780511525926

  20. Cahen, P.-J., Chabert, J.-L.: Integer-valued polynomials. Math. Surv. Monogr. (AMS) 48 (1997)

    Google Scholar 

  21. Dummit D.S., Foote R.M.: Abstract Algebra, 3rd ed. Wiley (2004)

    Google Scholar 

  22. Perez-Garcia, C., Schikhof, W. H.: Locally Convex Spaces over Non-Archimedean Valued Fields, pp. 472. Cambridge University Press (2010). https://doi.org/10.1017/CBO9780511729959

  23. Koblitz, N.: \(p\)-adic Numbers, \(p\)-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer, New York, (1977)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computer Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine

    Sergey Gefter, Anna Goncharuk & Aleksey Piven’

Authors
  1. Sergey Gefter
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  2. Anna Goncharuk
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  3. Aleksey Piven’
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Corresponding author

Correspondence to Anna Goncharuk .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Gefter, S., Goncharuk, A., Piven’, A. (2023). Implicit Linear First Order Difference Equations Over Commutative Rings. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_10

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