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Constructing universal Gröbner bases

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1987)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 356))

Abstract

A universal Gröbner basis is a finite basis for a polynomial ideal that has the Gröbner property with respect to all admissible term-orders. Let R be a commutative polynomial ring over a field K, or more generally a non-commutative polynomial ring of solvable type over K (see [KRW]). We show, how to construct and characterize left, right, two-sided, and reduced universal Gröbner bases in R. Moreover, we extend the upper complexity bounds in [We4] to the construction of universal Gröbner bases. Finally, we prove the stability of universal Gröbner bases under specialization of coefficients. All these results have counterparts for polynomial rings over commutative regular rings (comp. [We3]).

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References

  1. Proc. AAECC-4, Karlsruhe, 1986, Th. Beth, M. Clausen Eds., Springer LNCS vol. 307.

    Google Scholar 

  2. J. Apel, W. Lassner, An extension of Buchberger's algorithm and calculations in enveloping fields of Lie algebras, preprint 1985.

    Google Scholar 

  3. B.Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal, Dissertation, Universität Innsbruck, 1965.

    Google Scholar 

  4. B.Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, chap. 6 in Recent Trends in Multidimensional System Theory, N. K. Bose Ed., Reidel Publ. Comp., 1985.

    Google Scholar 

  5. Computer Algebra, Symbolic and Algebraic Computation, B.Buchberger, G. E.Collins, R. Loos Eds., Springer Verlag 1982/83.

    Google Scholar 

  6. EUROCAL '85, B.F.Caviness Ed., Springer LNCS vol. 204.

    Google Scholar 

  7. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York, printing 1984.

    Google Scholar 

  8. P. Gianni, Properties of Gröbner bases under specialization, presented at EUROCAL '87, Leipzig, June 1987.

    Google Scholar 

  9. M. Kalkbrener, Solving systems of algebraic equations by using Gröbner bases, presented at EUROCAL '87, Leipzig, June 1987.

    Google Scholar 

  10. A. Kandri-Rody, V.Weispfenning, Non-commutative Gröbner bases in algebras of solvable type, 1986, to appaer in J. Symb. Comp..

    Google Scholar 

  11. J. Keisler, Fundamentals of model theory, in Handbook of Mathematical Logic, J. Barwise Ed., North-Holland, 1977.

    Google Scholar 

  12. A. Levy, Basic Set Theory, Springer, 1979.

    Google Scholar 

  13. H. M. Möller, F. Mora, New constructive methods in classical ideal theory, J. of Algebra 100 (1986), 138–178.

    Article  Google Scholar 

  14. F. Mora, Gröbner bases for non-commutative polynomial rings, in AAECC-3, Grenoble 1985, Springer LNCS vol. 229, pp. 353–362.

    Google Scholar 

  15. Standard bases and non-Noetherianity: Non-commutative polynomial rings, in [AE4], pp. 98–109.

    Google Scholar 

  16. F. Mora, L. Robbiano, The Gröbner fan of an ideal, preprint, Universita di Genova, Feb. 1987.

    Google Scholar 

  17. M. O. Rabin, Decidable theories, in Handbook of Mathematical Logic, J. Barwise Ed., North-Holland, 1977.

    Google Scholar 

  18. L. Robbiano, Term orderings on the polynomial ring, in [EC85], pp. 513–517.

    Google Scholar 

  19. D. Saracino, V. Weispfenning, On algebraic curves over commutative regular rings, in Model Theory and Algebra, a memorial tribute to A. Robinson, Springer LNM vol. 489, pp. 307–383.

    Google Scholar 

  20. K.-P. Schemmel, An extension of Buchberger's algorithm to compute all reduced Gröbner bases of a polynomial ideal, presented at EUROCAL '87, Leipzig, June 1987.

    Google Scholar 

  21. N. Schwartz, Stability of Gröbner bases, J. pure and appl. Algebra 53 (1988), 171–186.

    Article  Google Scholar 

  22. J. R. Shoenfield, Mathematical Logic, Addison-Wesley, 1967.

    Google Scholar 

  23. V. Weispfenning, Model-completeness and elimination of quantifiers for subdirect products of structures, J. of Algebra, 36 (1975), 252–277.

    Article  Google Scholar 

  24. V. Weispfenning, Admissible orders and linear forms, ACM SIGSAM Bulletin 21,2 (1987), 16–18.

    Google Scholar 

  25. V. Weispfenning, Gröbner bases in polynomial rings over commutative regular rings, to appear in Proc. EUROCAL '87, Leipzig, Springer LNCS.

    Google Scholar 

  26. V. Weispfenning, Some bounds for the construction of Gröbner bases, 1987, in [AE4], pp. 195–201.

    Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl für Mathematik, Universität Passau, D-8390, Passau, FRG

    Volker Weispfenning

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  1. Volker Weispfenning
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Llorenç Huguet Alain Poli

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© 1989 Springer-Verlag Berlin Heidelberg

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Weispfenning, V. (1989). Constructing universal Gröbner bases. In: Huguet, L., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1987. Lecture Notes in Computer Science, vol 356. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51082-6_96

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  • DOI: https://doi.org/10.1007/3-540-51082-6_96

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51082-6

  • Online ISBN: 978-3-540-46150-0

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