Skip to main content
Log in

A Short Contribution to the Theory of Regular Chains

Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

This paper contains short contributions to the theory of regular chains which follow a recent JSC paper by the same authors. These contributions apply to both the nondifferential and the differential context. They deal with the computation of normal forms and with the membership problem to ideals defined by regular chains.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)

    Article  MathSciNet  Google Scholar 

  2. Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)

    Book  Google Scholar 

  3. Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the radical of a finitely generated differential ideal. In: ISSAC’95: Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, pp. 158–166. ACM Press, New York. http://hal.archives-ouvertes.fr/hal-00138020 (1995)

  4. Boulier, F., Lemaire, F.: A normal form algorithm for regular differential chains. Math. Comput. Sci. 4(2), 185–201 (2010). https://doi.org/10.1007/s11786-010-0060-3

    Article  MathSciNet  MATH  Google Scholar 

  5. Boulier, F., Lemaire, F., Poteaux, A., Moreno Maza, M.: An equivalence theorem for regular differential chains. J. Symb. Comput. 93, 34–55 (2019)

    Article  MathSciNet  Google Scholar 

  6. Boulier, F., Lemaire, F., Sedoglavic, A.: On the regularity property of differential polynomials modulo regular differential chains. In: Proceedings of Computer Algebra in Scientific Computing, LNCS 6885, pp. 61–72. Kassel, Germany. http://hal.archives-ouvertes.fr/hal-00599440 (2011)

  7. Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive triangular decompositions. In: Proceedings of CASC’07, pp. 73–101 (2007)

  8. Ducos, L.: Source of the axiom package prs.spad. http://www-math.sp2mi.univ-poitiers.fr/~ducos/src/travaux.html (1999)

  9. Hubert, É.: Notes on triangular sets and triangulation-decomposition algorithm I: Polynomial systems. Symb. Numer. Sci. Comput. 2001, 143–158 (2003)

    Google Scholar 

  10. Hubert, É.: Notes on triangular sets and triangulation-decomposition algorithm II: Differential systems. Symb. Numer. Sci. Comput. 2001, 40–87 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)

    MATH  Google Scholar 

  12. Morrison, S.: The differential ideal \([P] : M^\infty \). J. Symb. Comput. 28, 631–656 (1999)

    Article  Google Scholar 

  13. Ritt, J.F.: Differential Algebra, vol. 33. American Mathematical Society Colloquium Publications, American Mathematical Society, New York (1950)

    MATH  Google Scholar 

  14. Rosenfeld, A.: Specializations in differential algebra. Trans. Am. Math. Soc. 90, 394–407 (1959)

    Article  MathSciNet  Google Scholar 

  15. Wang, D.: Elimination Methods. Springer, Wien (2001)

    Book  Google Scholar 

  16. Yang, L., Zhang, J.: Searching dependency between algebraic equations: an algorithm applied to automated reasoning. In: Artificial Intelligence in Mathematics, pp. 147–156. 1991 version available at http://streaming.ictp.trieste.it/preprints/P/91/006.pdf (1994)

  17. Zariski, O., Samuel, P.: Commutative Algebra. Van Nostrand, New York. Also volumes 28 and 29 of the Graduate Texts in Mathematics. Springer, Berlin (1958)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to François Boulier.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by the bilateral Project ANR-17-CE40-0036 and DFG-391322026 SYMBIONT.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boulier, F., Lemaire, F., Moreno Maza, M. et al. A Short Contribution to the Theory of Regular Chains. Math.Comput.Sci. 15, 177–188 (2021). https://doi.org/10.1007/s11786-020-00477-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-020-00477-x

Keywords

Mathematics Subject Classification

Navigation