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Groebner basis

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Algebraic Geodesy and Geoinformatics

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Abstract

This chapter presents you the reader with one of the most powerful computer algebra tools, besides the polynomial resultants (discussed in the next chapter), for solving nonlinear systems of equations which you may encounter. The basic tools that you will require to develop your own algorithms for solving problems requiring closed form (exact) solutions are presented. This powerful tool is the “Gröbner basis” written in English as Groebner basis. It was first suggested by B. Buchberger in 1965, a PhD student of Wolfgang Groebner (1899 - 1980). Groebner, already in 1949, had suggested a method for finding a linearly independent basis of the vector space of the residue class ring of the polynomial ring modulo a polynomial ideal. In studying termination of this method, Buchberger came up both with the notion of Groebner bases (certain generating sets for polynomial ideals) and with an always terminating algorithm for computing them. In 1964, H. Hironaka (1931-) had independently introduced an analogous notion for the domain of power series in connection with his work on resolution of singularities in algebraic geometry and named it standard basis [262, p. 187]. However, he did not give any method for computing these bases.

“There are no good, general methods for solving systems of more than one nonlinear equation. Furthermore, it is not hard to see why (very likely) there never will be any good, general methods:...” W. H. Press et al.

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Authors and Affiliations

  1. Department of Spatial Sciences, Faculty of Science and Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia

    Prof. Dr.-Ing. Joseph L. Awange

  2. Geodätisches Institut Universität Stuttgart, Geschwister-Scholl-Str. 24 D, 70174, Stuttgart, Germany

    Prof. Dr.-Ing. Erik W. Grafarend

  3. Dept. Photogrammetry & Geoinformatics, Budapest University of Technology & Economics, 1521, Budapest, Hungary

    Prof.Dr. Béla Paláncz

  4. Dept. Geodesy & Surveying, Budapest University of Technology & Economics, Müegyetem rkp. 3, 1111, Budapest, Hungary

    Dr. Piroska Zaletnyik

Authors
  1. Prof. Dr.-Ing. Joseph L. Awange
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  2. Prof. Dr.-Ing. Erik W. Grafarend
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  3. Prof.Dr. Béla Paláncz
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  4. Dr. Piroska Zaletnyik
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Correspondence to Joseph L. Awange .

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Awange, J.L., Grafarend, E.W., Paláncz, B., Zaletnyik, P. (2010). Groebner basis. In: Algebraic Geodesy and Geoinformatics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12124-1_4

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