Advertisement

Algebraic factoring and geometry theorem proving

  • Dongming Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 814)

Abstract

Two methods for polynomial factorization over algebraic extension fields are reviewed. It is explained how geometric theorems may be proved by using irreducible zero decomposition for which algebraic factoring is necessary. A set of selected geometric theorems are taken as examples to illustrate how algebraic factoring can help understand the ambiguity of a theorem and prove it even if its algebraic formulation does not precisely correspond to the geometric statement. Among the polynomials occurring in our examples which need to be algebraically factorized, 12 are presented. Experiments with the two factoring methods for these polynomials are reported in comparison with the Maple's built-in factorizer. Our methods are always faster and any of the 12 polynomials can be factorized within 40 CPU seconds on a SUN SparcServer 690/51. Timings for proving the example theorems are also provided.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abbott, J. A.: On the factorization of polynomials over algebraic fields. Ph.D thesis. School of Math. Scis., Univ. of Bath, England (1989).Google Scholar
  2. [2]
    Chou, S. C.: Proving elementary geometry theorems using Wu's algorithm. In: Automated theorem proving: after 25 years, Contemp. Math. 29 (1984) 243–286.Google Scholar
  3. [3]
    Chou, S. C.: Mechanical geometry theorem proving. D. Reidel Publ. Co., Dordrecht-Boston-Lancaster-Tokyo (1988).Google Scholar
  4. [4]
    Chou, S. C., Gao, X. S.: Ritt-Wu's decomposition algorithm and geometry theorem proving. In: Proc. CADE-10, Lecture Notes in Comput. Sci. 449 (1990) 207–220.Google Scholar
  5. [5]
    Gao, X. S.: Transcendental functions and mechanical theorem proving in elementary geometries. J. Automat. Reason. 6 (1990) 403–417.Google Scholar
  6. [6]
    Hu, S., Wang, D. M.: Fast factorization of polynomials over rational number field or its extension fields. Kexue Tongbao 31 (1986) 150–156.Google Scholar
  7. [7]
    Landau, S.: Factoring polynomials over algebraic number fields. SIAM J. Comput. 14 (1985) 184–195.Google Scholar
  8. [8]
    Lenstra, A. K.: Lattices and factorization of polynomials over algebraic number fields. In: Proc. EUROCAM'82, Marseille (1982) 32–39.Google Scholar
  9. [9]
    Lenstra, A. K.: Factoring multivariate polynomials over algebraic number fields. SIAM J. Comput. 16 (1987) 591–598.Google Scholar
  10. [10]
    Ritt, J. F.: Differential algebra. New York: Amer. Math. Soc. (1950).Google Scholar
  11. [11]
    Trager, B. M.: Algebraic factoring and rational function integration: In: Proc. ACM Symp. Symb. Algebraic Comput., Yorktown Heights (1976) 219–226.Google Scholar
  12. [12]
    Wang, D. M.: Mechanical approach for polynomial set and its related fields (in Chinese). Ph.D thesis. Academia Sinica, China (1987).Google Scholar
  13. [13]
    Wang, D. M.: On Wu's method for proving constructive geometric theorems. In: Proc. IJCAI-89, Detroit (1989) 419–424.Google Scholar
  14. [14]
    Wang, D. M.: Characteristic sets and zero structure of polynomial sets. Lecture notes. RISC-LINZ, Joh Kepler Univ., Austria (1989).Google Scholar
  15. [15]
    Wang, D. M.: Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Comput. Aided Geom. Design 9 (1992) 471–484.Google Scholar
  16. [16]
    Wang, D. M.: A method for factorizing multivariate polynomials over successive algebraic extension fields. Preprint. RISC-LINZ, Joh Kepler Univ., Austria (1992).Google Scholar
  17. [17]
    Wang, D. M.: An elimination method for polynomial systems. J. Symb. Comput. 16 (1993) 83–114.Google Scholar
  18. [18]
    Wang, P. S.: Factoring multivariate polynomials over algebraic number fields. Math. Comput. 30 (1976) 324–336.Google Scholar
  19. [19]
    Weinberger, P. J., Rothschild, L. P.: Factoring polynomials over algebraic number fields. ACM Trans. Math. Software 2 (1976) 335–350.Google Scholar
  20. [20]
    Wu, W. T.: Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4 (1984) 207–235.Google Scholar
  21. [21]
    Wu, W. T.: Basic principles of mechanical theorem proving in geometries (Part on elementary geometries, in Chinese). Beijing: Science Press (1984).Google Scholar
  22. [22]
    Wu, W. T.: On reducibility problem in mechanical theorem proving of elementary geometries. Chinese Quart. J. Math. 2 (1987) 1–20.Google Scholar
  23. [23]
    Wu, W. T., Lu, X. L.: Triangles with equal bisectors (in Chinese). Beijing: People's Education Press (1985).Google Scholar
  24. [24]
    Yang, L., Zhang, J. Z.: Searching dependency between algebraic equations: an algorithm applied to automated reasoning. MM Res. Preprints 7 (1992) 105–114.Google Scholar
  25. [25]
    Yang, L., Zhang, J. Z., Hou, X. R.: An efficient decomposition algorithm for geometry theorem proving without factorization. MM Res. Preprints 9 (1993) 115–131.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Dongming Wang
    • 1
  1. 1.Laboratoire d'Informatique Fondamentale et d'Intelligence ArtificielleInstitut IMAGGrenoble CédexFrance

Personalised recommendations