Algorithmica

, Volume 5, Issue 1–4, pp 147–154 | Cite as

An algorithm for constructing gröbner bases from characteristic sets and its application to geometry

  • Shang-Ching Chou
  • William F. Schelter
  • Jin-Gen Yang
Article

Abstract

In Ritt's method, a prime ideal is given by a characteristic set. A characteristic set of a prime ideal is generally not a set of generators of this ideal. In this paper we present a simple algorithm for constructing Gröbner bases of a prime ideal from its characteristic set. We give a method for finding new theorems in geometry as an application of this algorithm.

Key words

Polynomial, (Prime) ideal Generators (Irreducible) ascending chain (Irreducible) algebraic set Decomposition of an algebraic set Geometric configuration Nondegenerate component Geometry theorem proving 

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References

  1. [1]
    B. Buchberger, Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Chapter 6 inRecent Trends in Multidimensional Systems Theory, N. K. Bose (ed.), Reidel, Dordrecht, 1985.Google Scholar
  2. [2]
    S.-C. Chou,Mechanical Geometry Theorem Proving, Reidel, Dordrecht, 1988.MATHGoogle Scholar
  3. [3]
    S.-C. Chou and G.-J. Yang, On the Algebraic Formulation of Certain Geometry Statements and Mechanical Geometry Theorem Proving, Preprint, May, 1986, revised in July, 1987, to appear inAlgorithmica. Google Scholar
  4. [4]
    P. Gianni, B. Trager, and G. Zacharias, Gröbner Bases and Primary Decomposition of Polynomial Ideals, Preprint, February, 1986.Google Scholar
  5. [5]
    R. F. Ritt,Differential Algebra, AMS Colloquium Publications, American Mathematical Society, Providence, RI, 1950.MATHGoogle Scholar
  6. [6]
    A. K. Rody, Effective Methods in the Theory of Polynomial Ideals, Ph.D. Thesis, Departments of Mathematical Sciences, Rensselaer Polytechnic Institute, 1984.Google Scholar
  7. [7]
    A. Seidenberg, Constructions in Algebra,Trans. Amer. Math. Soc.,197 (1974), 273–313.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Wu Wen-tsün, Basic Principles of Mechanical Theorem Proving in Geometries,J. Systems Sci Math. Sci,4(3), 1984, 207–235, republished inJ. Automated Reasoning,2(4) (1986), 221–252.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Shang-Ching Chou
    • 1
  • William F. Schelter
    • 2
  • Jin-Gen Yang
    • 2
  1. 1.Institute for Computing ScienceUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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