Skip to main content

Linear Differential Elimination for Analytic Functions

Mathematics in Computer Science volume 4, pages 231–242 (2010)Cite this article

Abstract

This paper provides methods to decide whether a given analytic function of several complex variables is a linear combination of finitely many given analytic functions with coefficients of the following special form: Each one of these coefficients is a composition of an unknown analytic function of less arguments than the function to be expressed, with fixed analytic functions. Methods which compute suitable coefficient functions in the affirmative case are presented as well.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The MAPLE Package “Janet”: I. Polynomial systems. II. Linear partial differential equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Proc. of Computer Algebra in Scientific Computing CASC 2003, pp. 31–40; 41–54, resp. Institut für Informatik, TU München, Garching, Germany (2003). Also available together with the package from http://wwwb.math.rwth-aachen.de/Janet

  2. Blinkov, Y.A., Gerdt, V.P., Yanovich, D.A.: Construction of Janet bases, II. Polynomial bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing CASC 2001, pp. 249–263. Springer, Berlin (2001)

  3. Cluzeau T., Quadrat A.: Factoring and decomposing a class of linear functional systems. Linear Algebra Appl. 428(1), 324–381 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Janet M.: Leçons sur les systèmes d’équations aux dérivées partielles, Cahiers Scientifiques IV. Gauthiers-Villars, Paris (1929)

    Google Scholar 

  5. Kolchin E.R.: Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, vol. 54. Academic Press, New York, London (1973)

    Google Scholar 

  6. Plesken W., Robertz D.: Janet’s approach to presentations and resolutions for polynomials and linear pdes. Arch. Math. 84(1), 22–37 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Plesken W., Robertz D.: Elimination for coefficients of special characteristic polynomials. Experiment. Math. 17(4), 499–510 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Thomas, J.M.: Differential Systems, vol. XXI. AMS Colloquium Publications (1937)

  9. Wolf T., Brand A., Mohammadzadeh M.: Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions. J. Symbol. Comput. 27(2), 221–238 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wolf, T.: Applications of CRACK in the classification of integrable systems. In: Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, vol. 37, pp. 283–300. Am. Math. Soc., Providence (2004)

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl B für Mathematik, RWTH Aachen University, Templergraben 64, 52062, Aachen, Germany

    Wilhelm Plesken & Daniel Robertz

Authors
  1. Wilhelm Plesken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Robertz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Robertz.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Plesken, W., Robertz, D. Linear Differential Elimination for Analytic Functions. Math.Comput.Sci. 4, 231–242 (2010). https://doi.org/10.1007/s11786-010-0053-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-010-0053-2

Keywords

  • Analytic function
  • Differential elimination
  • Functional dependence
  • Linear pde’s
  • Janet basis

Mathematics Subject Classification (2010)

  • Primary 12H05
  • Secondary 13N10