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Properties of Gröbner bases under specializations

  • Polynomial Algorithms
  • Conference paper
  • First Online:
Eurocal '87 (EUROCAL 1987)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 378))

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Abstract

In this paper we prove some properties of Gröbner bases under specialization maps. In particular we state sufficient conditions for the image of a Gröbner basis to be a Gröbner basis. We apply these results to the resolution of systems of polynomial equations. In particular we show that, if the system has a finite number of solutions, (in an algebraic closure of the base field K), the problem is totally reduced to a single Gröbner basis computation (w.r.t. purely lexicographical ordering), followed by a search for the roots of univariate polynomials and a “few” evaluations in suitable algebraic extensions of K.

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References

  1. B.Buchberger, “Gröbner bases: an algorithmic method in polynomial ideal theory”, in N.K.Bose (ed.): Recent trends in multidimensional systems theory, D.Rheidel Publ. Comp., Chapter 6.

    Google Scholar 

  2. P.Gianni, B.Trager, G.Zacharias, “Gröbner bases and primary decomposition of polynomial ideals”, to appear in J.of Symb. Comp..

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  3. Kalkbrener, “Solving Systems of Algebraic Equations by using Gröbner Bases”, In this Volume.

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  4. D.Lazard, “Ideal Bases and Primary Decomposition: case of two variables”, J.of Symb. Comp., Vol.1, n.3, September '85.

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  5. W.Trinks, “Über B.Buchbergers Verfharen, Systeme algebraisher Gleichungen zu losen, J.of Number Theory 10,pp. 475–488.

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Authors and Affiliations

  1. University of Pisa, Italy

    Patrizia Gianni

Authors
  1. Patrizia Gianni
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James H. Davenport

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© 1989 Springer-Verlag Berlin Heidelberg

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Gianni, P. (1989). Properties of Gröbner bases under specializations. In: Davenport, J.H. (eds) Eurocal '87. EUROCAL 1987. Lecture Notes in Computer Science, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51517-8_128

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  • DOI: https://doi.org/10.1007/3-540-51517-8_128

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51517-3

  • Online ISBN: 978-3-540-48207-9

  • eBook Packages: Springer Book Archive

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