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Homogenity, pseudo-homogenity, and Gröbner basis computations

  • Thomas Becker
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 539)

Abstract

Let K[X] be a multivariate polynomial ring over a field K, and let U ⊑ X. We call fK[X] pseudo-homogeneous in U if either it contains no variable of U, or each of its terms does. U ⊑ X is called an H-set for FK[X] if each fF is pseudo-homogeneous in U. We show that when computing an elimination ideal of some ideal (F) of K[X] w.r.t. V ⊑ X, one may disregard all those elements of F that contain variables from some H-set U ⊑ X/V for F. For given F and V, we compute a maximal H-set for F contained in V. Furthermore, we discuss how one can compute gradings by weighted total degree that make a given finite FK[X] homogeneous. For certain limited purposes, one can then compute truncated Gröbner bases instead of full ones.

Keywords

Integer Linear Programming Finite Subset Critical Pair Finite Basis Integer Linear Programming Problem 
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.

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References

  1. [1]
    W. Böge, R. Gebauer, and H. Kredel, Some examples for solving systems of algebraic equations by calculating Gröbner Bases, J. Symbolic Computation 1, 83–98 (1985)Google Scholar
  2. [2]
    B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory, in: N.K. Bose (ed.), Progress, Directions, and Open Problems in Systems Theory, 184–232, D. Riedel Publ. Comp. (Dordrecht 1985)Google Scholar
  3. [3]
    M. Caboara, Dynamic evaluation of Gr-bner bases, Communication at the CAMASA '91 Meeting, Cagliari 1991Google Scholar
  4. [4]
    E.G. Coffman Jr. (ed.), Computer and Job-Shop Scheduling Theory, John Wiley & Sons, New York (1976)Google Scholar
  5. [5]
    R. Gebauer, H.M. Möller, On an installation of Buchberger's algorithm, JSC 6 2/3, 185–91 (1988)Google Scholar
  6. [6]
    V.P. Gerdt, A.B. Shavachka, A. Zharkov, Computer algebra application for classification of integral non-linear evolution equation, JSC 1, 101–107 (1985)Google Scholar
  7. [7]
    P. Gritzman and B. Sturmfels, Minkowski addition of polytopes: computaional complexity and applications to Groebner bases, Preprint, Cornell University (1990)Google Scholar
  8. [8]
    D. Lazard, Gröbner bases, Gaussian elimination, and resolution of systems of algebraic equations, Proc. EUROCAL '83, Springer LNCS 162, 146–156 (1983)Google Scholar
  9. [9]
    C.E. Miller, A.W. Tucker, and R.A. Zemlin, Integer programming formulation and traveling salesman problems, J. ACM 7 (1960)Google Scholar
  10. [10]
    H.M. Moeller and T. Mora, Upper and lower bounds for the degree of Gröbner bases, Proc. EUROSAM '84, Springer LNCS 174, 172–183 (1984)Google Scholar
  11. [11]
    T. Mora and L. Robbiano, The Groebner fan of an ideal, JSC 6 183–208 (1988)Google Scholar
  12. [12]
    C.H. Papadimitrou and K. Steglitz, Combinatorical Optimization, Prentice Hall, Eaglewood Cliffs (1982)Google Scholar
  13. [13]
    L. Robbiano, On the theory of graded structures, JSC 2, 139–170 (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Thomas Becker
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität PassauPassauFR Germany

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