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Differential and Difference Dimension Polynomials

  • Book
  • © 1999

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Authors:
  1. M. V. Kondratieva

    1. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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  2. A. B. Levin

    1. Department of Mathematics, The Catholic University of America, USA

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  3. A. V. Mikhalev

    1. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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  4. E. V. Pankratiev

    1. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 461)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Preliminaries

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 1-44

  3. Numerical Polynomials

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 45-121

  4. Basic Notions of Differential and Difference Algebra

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 123-190

  5. Gröbner Bases

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 191-221

  6. Differential Dimension Polynomials

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 223-280

  7. Dimension Polynomials in Difference and Difference-Differential Algebra

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 281-353

  8. Some Application of Dimension Polynomials in Difference-Differential Algebra

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 355-375

  9. Dimension Polynomials of Filtered G-Modules and Finitely Generated G-Fields Extensions

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 377-396

  10. Computation of Dimension Polynomials

    • M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev
    Pages 397-403

  11. Back Matter

    Pages 405-426
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Keywords

About this book

The role of Hilbert polynomials in commutative and homological algebra as well as in algebraic geometry and combinatorics is well known. A similar role in differential algebra is played by the differential dimension polynomials. The notion of differential dimension polynomial was introduced by E. Kolchin in 1964 [KoI64]' but the problems and ideas that had led to this notion (and that are reflected in this book) have essentially more long history. Actually, one can say that the differential dimension polynomial describes in exact terms the freedom degree of a dynamic system as well as the number of arbitrary constants in the general solution of a system of algebraic differential equations. The first attempts of such description were made at the end of 19th century by Jacobi [Ja890] who estimated the number of algebraically independent constants in the general solution of a system of linear ordinary differential equations. Later on, Jacobi's results were extended to some cases of nonlinear systems, but in general case the problem of such estimation (that is known as the problem of Jacobi's bound) remains open. There are some generalization of the problem of Jacobi's bound to the partial differential equations, but the results in this area are just appearing. At the beginning of the 20th century algebraic methods in the theory of differen­ tial equations were actively developed by F. Riquier [RiqlO] and M.

Authors and Affiliations

  • Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

    M. V. Kondratieva, A. V. Mikhalev, E. V. Pankratiev

  • Department of Mathematics, The Catholic University of America, USA

    A. B. Levin

Bibliographic Information

  • Book Title: Differential and Difference Dimension Polynomials

  • Authors: M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev

  • Series Title: Mathematics and Its Applications

  • DOI: https://doi.org/10.1007/978-94-017-1257-6

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 1999

  • Hardcover ISBN: 978-0-7923-5484-0Published: 30 November 1998

  • Softcover ISBN: 978-90-481-5141-7Published: 06 December 2010

  • eBook ISBN: 978-94-017-1257-6Published: 09 March 2013

  • Edition Number: 1

  • Number of Pages: XIII, 422

  • Topics: Algebra, Partial Differential Equations, Combinatorics

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