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A note on the complexity of constructing Gröbner-bases

  • Algorithms 2 — Polynomial Ideal Bases
  • Conference paper
  • First Online:
Book cover Computer Algebra (EUROCAL 1983)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 162))

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Abstract

In the bivariate case, upper bounds for the degrees and the number of polynomials occuring in Gröbner-bases of polynomial ideals are given. In the case of the total degree ordering of monomials, the upper bound for the degrees is linear in the maximal degree of the polynomials in the given basis of the ideal. In the general case, the upper bound for the degrees is quadratic. The upper bound for the number of polynomials is linear in the minimal degree of the polynomials in the given basis. All the bounds are shown to be tight. The relevance of these bounds for constructive polynomial ideal theory is indicated.

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References

  • Bayer, D.A., 82: The Division Algorithm and the Hilbert Scheme. Harvard University, Cambridge, Mass., Math. Dptmt: Ph.D. Thesis, June 1982.

    Google Scholar 

  • Buchberger, B., 65: An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal (German). Univ. of Innsbruck, Austria: Math. Inst., Ph.D. Thesis 1965.

    Google Scholar 

  • Buchberger, B., 70: An Algorithmical Criterion for the Sovability of Algebraic Systems of Equations (German). Aequationes mathematicae 4/3, 374–383 (1970).

    Google Scholar 

  • Buchberger, B., 76a: A Theoretical Basis for the Reduction of Polynomials to Canonical Form. ACM SIGSAM Bull. 10/3, 19–29 (1976).

    Google Scholar 

  • Buchberger, B., 76b: Some Properties of Gröbner-Bases for Polynomial Ideals. ACM SIGSAM Bull. 10/4, 19–24 (1976).

    Google Scholar 

  • Buchberger, B., 79: A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner-Bases. Proc. EUROSAM 1979, Marseille, Lect. Notes Comput. Sci. 72, 3–21 (1979).

    Google Scholar 

  • Buchberger, B., 82: Miscellaneous Results on Gröbner-Bases for Polynomial Ideals II. Technical Report, Dpmt. of Computer and Information Sciences, University of Delaware, to appear (1982).

    Google Scholar 

  • Buchberger, B., Loos, R., 82: Algebraic Simplification. In: Computer Algebra (B. Buchberger, G. Collins, R. Loos eds.), Springer, Wien-New York, 1982, 11–43.

    Google Scholar 

  • Buchberger, B., Winkler, F., 79: Miscellaneous Results on the Construction of Gröbner-Bases for Polynomial Ideals I. Technical Report Nr. 137, Mathematisches Institut, Universität Linz, Austria (1979).

    Google Scholar 

  • Cardoza, E., Lipton, R., Meyer, A.R., 76: Exponential Space Complete Problems for Petri Nets and Commutative Semigroups. Conf. Record of the 8th Annual ACM Symp. on Theory of Computing, 50–54 (1976).

    Google Scholar 

  • Guiver, J.P., 82: Contributions to Two-Dimensional System Theory. Univ. of Pittsburgh, Math. Depmt.: Ph.D. Thesis 1982.

    Google Scholar 

  • Hermann, G., 26: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95, 736–788 (1926).

    Google Scholar 

  • Lazard, D., 83: Gröbner bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations. These proceedings.

    Google Scholar 

  • Loos, R., 82: Generalized Polynomial Remainder Sequences. In: Computer Algebra (B. Buchberger, G. Collins, R. Loos eds.), Springer, Wien-New York, 1982, 115–137.

    Google Scholar 

  • Mayr, E.W., Meyer, A.R., 81: The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals. M.I.T.: Lab. Comput. Sci. Rep. LCS/TM-199 (1981).

    Google Scholar 

  • Pohst, M., Yun, D.Y.Y., 81: On Solving Systems of Algebraic Equations Via Ideal Bases and Elimination Theory SYMSAC 1981, 206–211.

    Google Scholar 

  • Sakata, S., 81: On Determining the Independent Point Set for Doubly Periodic Arrays and Encoding Two-Dimensional Cyclic Codes and Their Duals. IEEE Trans. on Information Theory, IT-27/5, 556–565.

    Google Scholar 

  • Schaller, S., 79: Algorithmic Aspects of Polynomial Residue Class Rings. University of Wisconsin, Madison: Ph.D. Thesis, Comput. Sci. Tech. Rep. 370, 1979.

    Google Scholar 

  • Seidenberg, A., 74: Constructions in Algebra. Trans. AMS 197, 273–313 (1974).

    Google Scholar 

  • Trinks, W., 78: On B. Buchberger's method of Solving Algebraic Equations (German). J. Number Theory 10/4, 475–488 (1978).

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Universität Linz, A4040, Linz, Austria

    B. Buchberger

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  1. B. Buchberger
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J. A. van Hulzen

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© 1983 Springer-Verlag

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Buchberger, B. (1983). A note on the complexity of constructing Gröbner-bases. In: van Hulzen, J.A. (eds) Computer Algebra. EUROCAL 1983. Lecture Notes in Computer Science, vol 162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12868-9_98

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  • DOI: https://doi.org/10.1007/3-540-12868-9_98

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12868-7

  • Online ISBN: 978-3-540-38756-5

  • eBook Packages: Springer Book Archive

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