Results in Mathematics

, Volume 15, Issue 3–4, pp 351–360 | Cite as

Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

  • Bernd Sturmfels


Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry.

In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.


Invariant Theory Symbolic Computation Zariski Closure Real Algebraic Geometry Fundamental Invariant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Becker, On the real spectrum of a ring and its applications to semialgebraic geometry, Bull. Amer. Math. Soc. 15 (1986) 19–60.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    J. Bokowski and B. Sturmfels, Polytopal and non-polytopal spheres. An algorithmic approach, Israel J. Math. 57 (1987) 257–271.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J. Bokowski and K. Garms, Altshuler’s sphere M425 10 is not polytopal, European J. Combinatorics 8 (1987) 227–229.MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Bokowski and B. Sturmfels, Computational Synthetic Geometry, Lecture Notes in Mathematics 1355, Springer, Heidelberg, 1989.Google Scholar
  5. 5.
    W.D. Brownawell, Bounds for the degree in the Nullstellensatz, Annals of Math. 126 (1987) 577–591.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystemes, Aequationes Mathematicae 4 (1970) 374–383.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    B. Buchberger, Gröbner bases — an algorithmic method in polynomial ideal theory, Chapter 6 in N.K. Bose (ed.): “Multidimensional Systems Theory”, D. Reidel Publ., 1985.Google Scholar
  8. 8.
    B. Buchberger, Applications of Gröbner bases in non-linear computational geometry, in J.R. Rice (ed.): Scientific Software, I.M.A. Volumes in Mathematics and its Applications, # 14, Springer, New York, 1988.Google Scholar
  9. 9.
    M.A. Dickrnann, Applications of model theory to real algebraic geometry, in “Methods in Mathematical Logic”, Lecture Notes in Mathematics 1130, Springer, Heidelberg, 1983, pp. 77–150.Google Scholar
  10. 10.
    J.A. Dieudonné and J.B. Carrell, Invariant Theory — Old and New, Academic Press, New York, 1971.zbMATHGoogle Scholar
  11. 11.
    D.Y. Grigor’ev and N.N. Vorobjov, Solving systems of polynomial inequalities in sub exponential time, J. Symbolic Computation 5 (1988) 37–64.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    G. Herrmann, Die Frage der endlich vielen Schritte in der Theory der Polynomideale, Math. Annalen 95 (1926) 736–788.CrossRefGoogle Scholar
  13. 13.
    C.R. Johnson, Some outstanding problems in the theory of matrices, Linear and Multilinear Algebra 12 (1982) 99–108.zbMATHCrossRefGoogle Scholar
  14. 14.
    D. Kapur, Using Gröbner bases to reason about geometry, J. Symbolic Computation 2 (1986) 399–408.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    L.M. Kelly and S. Nwankpa, Affine embeddings of Sylvester-Gallai designs, J. Combinatorial Theory 14 (1973) 422–438.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D. Lazard, Algébre linéaire sur K[X1,…,X n] et élimination, Bull. Soc. math. France 105 (1977) 165–190.MathSciNetzbMATHGoogle Scholar
  17. 17.
    T. McMillan and N. White, Cayley Factorization, in P. Gianni (ed.): Proc. ACM Intern. Symp. Symbolic and Algebraic Computation, Rome, July 1988, to appear.Google Scholar
  18. 18.
    D. Mumford, The Red Book on Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer, Heidelberg, 1988.Google Scholar
  19. 19.
    D. Shannon and M. Sweedler, Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms, and to determine birational equivalence, J. Symbolic Computaion 6 (1988) 267–274.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    B. Sturmfels and N. White, Gröbner bases and invariant theory, Advances in Math., to appear.Google Scholar
  21. 21.
    B.L. van der Waerden, Algebra I, Springer, Berlin, 1971.zbMATHGoogle Scholar
  22. 22.
    H. Weyl, The Classical Groups, Princeton University Press, Princeton, N.J., 1946.zbMATHGoogle Scholar
  23. 23.
    W. Whiteley, Logic and Invariant Theory, Ph.D. Dissertation, Massachussetts Institute of Technology, 1971.Google Scholar
  24. 24.
    W. Whiteley, Logic and invariant computation for analytic geometry, in “Symbolic Computations in Geometry”, I.M.A. Preprint # 389, University of Minnesota, January 1988.Google Scholar
  25. 25.
    W. Wu, Basic principles of mechanical theorem proving in geometries, J. of Automated Reasoning 2 (1986) 221–252.zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1989

Authors and Affiliations

  • Bernd Sturmfels
    • 1
  1. 1.Research Institute for Symbolic ComputationJohannes-Kepler-Universität LinzLinzAustria

Personalised recommendations