Skip to main content
SpringerLink
Log in

Notes on Triangular Sets and Triangulation-Decomposition Algorithms I: Polynomial Systems

  • Conference paper
  • First Online:
Symbolic and Numerical Scientific Computation (SNSC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2630))

Abstract

This is the first in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results. We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. We address here polynomial systems and some of the material we develop here will be used in the second part, devoted to differential systems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. A. Abramov and K. Y. Kvashenko. The greatest common divisor of polynomials that depend on a parameter. Vestnik Moskovskogo Universiteta. Seriya XV. Vychislitel’ naya Matematika i Kibernetika, 2: 65–71, 1993. translation in Moscow Univ. Comput. Math. Cybernet. 1993, no. 2, 59–64.

    MathSciNet  Google Scholar 

  2. P. Aubry, D. Lazard, and M. Moreno-Maza. On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2), 1999.

    Google Scholar 

  3. P. Aubry and M. Moreno-Maza. Triangular sets for solving polynomial systems: a comparative implementation of four methods. Journal of Symbolic Computation, 28(1–2):125–154, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Aubry, F. Rouillier, and M. Safey El Din. Real solving for positive dimensional systems. Journal of Symbolic Computation, 33(6):543–560, 2003.

    MathSciNet  Google Scholar 

  5. P. Aubry. Ensembles triangulaires de polynomes et résolution de systèmes algébriques. Implantation en Axiom. PhD thesis, Université de Paris 6, 1999

    Google Scholar 

  6. A. M. Bellido and V. Jalby. Spécifcations des algorithmes de triangulation de systèmes algébro-élémentaires. C. R. Acad. Sci. Paris, Ser. I(334):155–159, 2001.

    Google Scholar 

  7. D. Bouziane, A. Kandri Rody, and H. Maârouf. Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. Journal of Symbolic Computation, 31(6):631–649, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Boulier and F. Lemaire. Computing canonical representatives of regular differential ideals. In C. Traverso, editor, ISSAC. ACM-SIGSAM, ACM, 2000.

    Google Scholar 

  9. F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a finitely generated differential ideal. In A. H. M. Levelt, editor, ISSAC’95. ACM Press, New York, 1995.

    Google Scholar 

  10. F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Computing representations for radicals of finitely generated differential ideals. Technical Report IT-306, LIFL, 1997.

    Google Scholar 

  11. T. Becker and V. Weispfenning. Gröbner Bases — A Computational Approach to Commutative Algebra. Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  12. S. C. Chou and X. S. Gao. Ritt-wu’s decomposition algorithm and geometry theorem proving. In M. E. Stickel, editor, 10th International Conference on Automated Deduction, number 449 in Lecture Notes in Computer Sciences, pages 207–220. Springer-Verlag, 1990.

    Google Scholar 

  13. S. C. Chou, X. S. Gao, and J. Z. Zhang. Machine proofs in geometry. World Scientific Publishing Co. Inc., River Edge, NJ, 1994. Automated productionof readable proofs for geometry theorems, With a foreword by Robert S. Boyer.

    MATH  Google Scholar 

  14. T. Cluzeau and E. Hubert. Resolvent representation for regular differential ideals. Applicable Algebra in Engineering, Communication and Computing, 13(5):395–425, 2003.

    Article  MathSciNet  Google Scholar 

  15. S. C. Chou. Mechanical geometry theorem proving. D. Reidel Publishing Co., Dordrecht, 1988. With a foreword by Larry Wos.

    MATH  Google Scholar 

  16. J. Della Dora, C. Dicrescenzo, and D. Duval. About a new method for computing in algebraic number fields. In Proceedings of EUROCAL 85, number 204 in Lecture Notes in Computer Science, pages 289–290. Springer-Verlag, 1985.

    Google Scholar 

  17. S. Dellière. Triangularisation de systèmes constructibles. Application à l’évaluation dynamique. PhD thesis, Université de Limoges, 1999.

    Google Scholar 

  18. S. Dellière. D. M. Wang simple systems and dynamic constructible closure. Technical Report 16, Laboratoire d’Arithmétique, de Calcul Formel et d’Optimisation, Limoges, http://www.unilim.fr/laco, 2000.

    Google Scholar 

  19. S. Dellière. A first course to D7 with examples. http://www-lmc.imag.fr/lmc-cf/Claire.Dicrescenzo/D7, 2000.

  20. S. Dellière. Pgcd de deux polynômes à paramètres: approche par la clôture constructible dynamique et généralisation de la méthode de S. A. Abramov, K. Yu. Kvashenko. Technical Report RR-3882, INRIA, Sophia Antipolis, 2000. http://www.inria.fr/RRRT/RR-3882.html

    Google Scholar 

  21. S. Dellière. On the links between triangular sets and dynamic constructible closure. J. Pure Appl. Algebra, 163(1):49–68, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  22. D. Duval. Diverses questions relatives au calcul formel avec des nombres algébriques. PhD thesis, Institut Fourier, Grenoble, 1987.

    Google Scholar 

  23. D. Eisenbud. Commutative Algebra with a View toward Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag New York, 1994.

    Google Scholar 

  24. M. V. Foursov and M. Moreno Maza. On computer-assisted classification of coupled integrable equations. In Proceedings of the 2001 international symposium on Symbolic and algebraic computation, pages 129–136. ACM Press, 2001.

    Google Scholar 

  25. X. S. Gao and S. C. Chou. Solving parametric algebraic sytems. In ISSAC 92, pages 335–341. ACM Press, New York, NY, USA, 1992.

    Chapter  Google Scholar 

  26. K. O. Geddes, S. R. Czapor, and G. Labahn. Algorithms for computer algebra. Kluwer Academic Publishers, Boston, MA, 1992.

    MATH  Google Scholar 

  27. T. Gómez-Dîaz. Quelques applications de l’évaluation dynamique. PhD thesis, Université de Limoges, 1994.

    Google Scholar 

  28. E. Hubert. Factorisation free decomposition algorithms in differential algebra. Journal of Symbolic Computation, 29(4–5):641–662, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms. II Differential systems. In this volume, 2003.

    Google Scholar 

  30. M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation, 15(2):143–167, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  31. M. Kalkbrener. Algorithmic properties of polynomial rings. Journal of Symbolic Computation, 26(5):525–582, November 1998.

    Article  MATH  MathSciNet  Google Scholar 

  32. E. R. Kolchin. Differential Algebra and Algebraic Groups, volume 54 of Pure and Applied Mathematics. Academic Press, New York-London, 1973.

    MATH  Google Scholar 

  33. D. Lazard. A new method for solving algebraic systems of positive dimension. Discrete and Applied Mathematics, 33:147–160, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  34. D. Lazard. Solving zero dimensional algebraic systems. Journal of Symbolic Computation, 15: 117–132, 1992.

    Article  MathSciNet  Google Scholar 

  35. H. Maârouf, A. Kandri Rody, and M. Ssafini. Triviality and dimension of a system of algebraic differential equations. Journal of Automated Reasoning, 20(3):365–385, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  36. M. Moreno-Maza. Calculs de pgcd au-dessus des tours d’extensions simples et résolution des systèmes d’équations algébriques. PhD thesis, Université Paris 6, 1997.

    Google Scholar 

  37. M. Moreno Maza and R. Rioboo. Polynomial gcd computations over towers of algebraic extensions. In Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), pages 365–382. Springer, Berlin, 1995.

    Google Scholar 

  38. S. Morrison. The differential ideal [P]: M ∞. Journal of Symbolic Computation, 28(4–5):631–656, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  39. D. Richardson. Weak Wu stratification in R n. Journal of Symbolic Computation, 28(1–2):213–223, 1999. Polynomial elimination—algorithms and applications.

    Article  MATH  MathSciNet  Google Scholar 

  40. J. F. Ritt. Manifolds of functions defined by systems of algebraic differential equations. Transaction of the American Mathematical Society, 32:569–598, 1930.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. F. Ritt. Differential Equations from the Algebraic Standpoint. Amer. Math. Soc. Colloq. Publ., 1932.

    Google Scholar 

  42. J. F. Ritt. Differential Algebra, volume XXXIII of Colloquium publications. American Mathematical Society, 1950. Reprinted by Dover Publications, Inc (1966).

    Google Scholar 

  43. E. Schost. Sur la résolution des systèmes polynomiaux à paramètres. PhD thesis, École polytechnique, 2000.

    Google Scholar 

  44. J. Schicho and Z. Li. A construction of radical ideals in polynomial algebra. Technical Report 98-17, RISC-Linz, 1998. ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1998/98-17.ps.gz.

  45. A. Szanto. Computation with polynomial systems. PhD thesis, Cornell University, 1998.

    Google Scholar 

  46. W. V. Vasconcelos. Computational Methods in Commutative Algebra and Algebraic Geometry, volume 2 of Algorithms and Computation in Mathematics. Springer, Berlin, 1998.

    Google Scholar 

  47. J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, New York, 1999.

    MATH  Google Scholar 

  48. D. Wang. An elimination method for polynomial systems. Journal of Symbolic Computation, 16(2): 83–114, August 1993.

    Article  MATH  MathSciNet  Google Scholar 

  49. D. Wang. Decomposing polynomial systems into simple systems. Journal of Symbolic Computation, 25(3):295–314, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  50. D. Wang. Elimination methods. Texts and Monographs in Symbolic Computation. Springer-Verlag Wien, 1999.

    Google Scholar 

  51. D. Wang. Computing triangular systems and regular systems. Journal of Symbolic Computation, 30(2):221–236, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  52. W. T. Wu. On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci. Sinica, 21(2):159–172, 1978.

    MATH  MathSciNet  Google Scholar 

  53. W. T. Wu. Mechanical theorem proving in geometries. Springer-Verlag, Vienna, 1994. Basic principles, Translated from the 1984 Chinese original by Xiao Fan Jin and Dong Ming Wang.

    Google Scholar 

  54. W. T. Wu. Mathematics mechanization. Kluwer Academic Publishers Group, Dordrecht, 2000. Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA - Projet CAFE, 2004 route des lucioles, BP 93, F-06902, Sophia Antipolis

    Evelyne Hubert

Authors
  1. Evelyne Hubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RISC-Linz, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Franz Winkler

  2. Institute of Computational Mathematics, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Ulrich Langer

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hubert, E. (2003). Notes on Triangular Sets and Triangulation-Decomposition Algorithms I: Polynomial Systems. In: Winkler, F., Langer, U. (eds) Symbolic and Numerical Scientific Computation. SNSC 2001. Lecture Notes in Computer Science, vol 2630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45084-X_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-45084-X_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40554-2

  • Online ISBN: 978-3-540-45084-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation