IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-94-017-0337-6_1.html
   My bibliography  Save this book chapter

A Survey on Polynomials and Polynomial and Compatible Functions

In: Proceedings of the Third International Algebra Conference

Author

Listed:
  • Erhard Aichinger

    (Johannes Kepler University Linz, Department of Algebra, Stochastics, and Knowledge Based Mathematical Systems)

  • Günter F. Pilz

    (Johannes Kepler University Linz, Department of Algebra, Stochastics, and Knowledge Based Mathematical Systems)

Abstract

The notion of a polynomial over a commutative ring with unity is well-known. At first sight, however, it is not so clear how to define polynomials over non-commutative rings (do the “variables” have to commute with the coefficients?) and over general algebras. To every polynomial, there is a corresponding polynomial function. When is this correspondence one-to-one? Since the “variables” and the constants preserve all congruences, all polynomial functions do the same: they are “congruence compatible” functions. But when is every congruence compatible function a polynomial function? In section 1, we explain how to define polynomials and polynomial equations over general algebras, and we state several results on the solvability of such equations. In section 2, we study polynomial functions, and in section 3, we state the answers to some questions concerning polynomial and affine completeness. We focus on those results that are applicable to polynomial functions on groups, rings, vector-spaces, modules, or, more generally, on Ω-groups.

Suggested Citation

  • Erhard Aichinger & Günter F. Pilz, 2003. "A Survey on Polynomials and Polynomial and Compatible Functions," Springer Books, in: Proceedings of the Third International Algebra Conference, pages 1-16, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-0337-6_1
    DOI: 10.1007/978-94-017-0337-6_1
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-94-017-0337-6_1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.