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On an implementation of standard bases and syzygies in SINGULAR

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We describe an implementation of a general standard basis algorithm, valid for any monomial ordering compatible with the natural semigroup structure. We concentrate on new strategies which have proved useful, in particular in the non-wellordering case. Moreover, we describe the first implementation of Schreyer's method to compute syzygies and compare it with other, more classical methods. Comparisons of the computing time of several examples in the system SINGULAR give hints as to which strategies should be used in different classes of examples.

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References

  1. Bayer, D., Stillman, M.: Macaulay. A computer algebra system for algebraic geometry

  2. Becker, T., Weispfenning, V.: Gröbner Bases. A computational approach to commutative algebra. Berlin, Heidelberg, New York: Springer GTM 141 (1991)

    Google Scholar 

  3. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Thesis, Univ. Innsbruck, 1965

  4. Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory, In: Bose N.K. (ed) Multidimensional system theory, Riedel (1985)

  5. Eisenbud, D.: Commutative Algebra with a view toward Algebraic Geometry. New York, Heidelberg, Berlin: Springer 1995

    Google Scholar 

  6. Grabe, H.-G.: The tangent cone algorithm and homogenization. J. Pure Appl. Alg97, 303–312 (1994)

    Google Scholar 

  7. Gebauer, R., Möller, M.: On an installation of Buchberger's Algorithm. J. Symbolic Computation6, 275–286 (1988)

    Google Scholar 

  8. Giovini, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C.: “One sugar cube, please” or selection strategies in the Buchberger algorithm. Proceedings of the 1991 ISSAC, 55–63

  9. Greuel, G.-M., Pfister, G.: Advances and improvements in the theory of standard bases and syzygies. Archiv Math63 (1995)

  10. Lazard, D.: Gröbner bases, Gaussian elimination, and resolution of systems of algebraic equations. Proc. EUROCAL 83, LN Comp. Sci.162, 146–156

  11. Mora, T.: An algorithm to compute the equations of tangent cones. Proc. EUROCAM 82, Berlin, Heidelberg, New York: Springer, Lecture Notes in Computer Science (1982)

    Google Scholar 

  12. Mora, T.: Seven variations on standard bases. Preprint, Univ. Genova (1988)

  13. Möller, H. M., Mora, T.: Computational aspects of reduction strategies to construct resolutions of monomial ideals. Proc. AAECC 2, Lecture Notes in Computer Science228 (1986)

  14. Möller. H. M., Mora, T.: New Constructive Methods in Classical Ideal Theory. J. Algebra100, 138–178 (1986)

    Google Scholar 

  15. Möller, H. M., Mora, T., Traverso, C.: Gröbner bases computation using syzygies. Proc. of ISSAC 1992

  16. Mora, T., Pfister, G., Traverso, C.: An introduction to the tangent cone algorithm. Advances in Computing research, Issues in Robotics and nonlinear geometry6, 199–270 (1992)

    Google Scholar 

  17. Pohl, W.: About the weighted ecart and the weighted sugarMethod for computing standard bases. Preprint, Univ. Kaiserslautern, 1994

  18. Schreyer, F.-O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math.421, 83–123 (1991)

    Google Scholar 

  19. Zimnol, M.: Beispiele algebraischer Reduktionsstrukturen. Diplomarbeit, Kaiserslautern (1987)

Download references

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    G. -M. Greuel, G. Pfister, W. Pohl, H. Schönemann & T. Siebert

  2. Humboldt-Universität Berlin, Germany

    H. Grassmann

  3. Universität Cottbus, Germany

    B. Martin

  4. IBM Frankfurt, Germany

    W. Neumann

Authors
  1. H. Grassmann
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  2. G. -M. Greuel
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  3. B. Martin
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  4. W. Neumann
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  5. G. Pfister
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  6. W. Pohl
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  7. H. Schönemann
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  8. T. Siebert
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Grassmann, H., Greuel, G.M., Martin, B. et al. On an implementation of standard bases and syzygies in SINGULAR. AAECC 7, 235–249 (1996). https://doi.org/10.1007/BF01190332

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  • DOI: https://doi.org/10.1007/BF01190332

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