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Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One

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Computer Algebra in Scientific Computing (CASC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8136))

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Abstract

For a regular chain R in dimension one, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of R, that is, the set \(\overline{W(R)} \setminus W(R)\). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We provide experimental results illustrating the benefits of our algorithms.

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References

  1. Alonso, M.E., Mora, T., Niesi, G., Raimondo, M.: An algorithm for computing analytic branches of space curves at singular points. In: Proc. of the 1992 International Workshop on Mathematics Mechanization, pp. 135–166 (1992)

    Google Scholar 

  2. Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the radical of a finitely generated differential ideal. In: Proc. of ISSAC 1995, pp. 158–166 (1995)

    Google Scholar 

  3. Boulier, F., Lemaire, F., Moreno Maza, M.: Well known theorems on triangular systems and the D5 principle. In: Proc. of Transgressive Computing 2006, Granada, Spain, pp. 79–91 (2006)

    Google Scholar 

  4. Chen, C., Davenport, J.H., May, J.P., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. J. Symb. Comput. 49, 3–26 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C., Golubitsky, O., Lemaire, F., Maza, M.M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, C., Moreno Maza, M., Xia, B., Yang, L.: Computing cylindrical algebraic decomposition via triangular decomposition. In: Proc. of ISSAC 2009, pp. 95–102 (2009)

    Google Scholar 

  8. Chou, S.C., Gao, X.S.: A zero structure theorem for differential parametric systems. J. Symb. Comput. 16(6), 585–595 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Proc. of EUROCAL 1985, pp. 289–290 (1985)

    Google Scholar 

  10. Duval, D.: Rational Puiseux expansions. Compos. Math. 70(2), 119–154 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Fischer, G.: Plane Algebraic Curves. American Mathematical Society (2001)

    Google Scholar 

  12. Gao, X.S., Van der Hoeven, J., Yuan, C.M., Zhang, G.L.: Characteristic set method for differential-difference polynomial systems. J. Symb. Comput. 44(9), 1137–1163 (2009)

    Article  MATH  Google Scholar 

  13. Hubert, E.: Factorization-free decomposition algorithms in differential algebra. J. Symb. Comput. 29(4-5), 641–662 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kalkbrener, M.: Algorithmic properties of polynomial rings. J. Symb. Comput. 26(5), 525–581 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lemaire, F., Moreno Maza, M., Pan, W., Xie, Y.: When does 〈t 〉 equal sat(t)? J. Symb. Comput. 46(12), 1291–1305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library. In: Maple 10, Maplesoft, Canada (2005); refereed software

    Google Scholar 

  17. Marcus, S., Maza, M.M., Vrbik, P.: On fulton’s algorithm for computing intersection multiplicities. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 198–211. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Maurer, J.: Puiseux expansion for space curves. Manuscripta Math. 32, 91–100 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mumford, D.: The Red Book of Varieties and Schemes, 2nd edn. Springer (1999)

    Google Scholar 

  20. Munkres, J.R.: Topology, 2nd edn. Prentice Hall (2000)

    Google Scholar 

  21. Parusiński, A., Rond, G.: The Abhyankar-Jung theorem. J. Algebra 365, 29–41 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ritt, J.F.: Differential Equations from an Algebraic Standpoint, vol. 14. American Mathematical Society (1932)

    Google Scholar 

  23. Shimoyama, T., Yokoyama, K.: Localization and primary decomposition of polynomial ideals. J. Symb. Comput. 22(3), 247–277 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. van Hoeij, M.: An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comput. 18(4), 353–363 (1994)

    Article  MATH  Google Scholar 

  25. Walker, R.J.: Algebraic Curves. Springer (1978)

    Google Scholar 

  26. Wang, D.K.: The Wsolve package, http://www.mmrc.iss.ac.cn/~dwang/wsolve.html

  27. Wang, D.M.: Epsilon 0.618, http://www-calfor.lip6.fr/~wang/epsilon

  28. Yang, L., Hou, X.R., Xia, B.: A complete algorithm for automated discovering of a class of inequality-type theorems. Science in China, Series F 44(1), 33–49 (2001)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. ORCCA, University of Western Ontario (UWO), London, Ontario, Canada

    Parisa Alvandi, Changbo Chen & Marc Moreno Maza

Authors
  1. Parisa Alvandi
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  2. Changbo Chen
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  3. Marc Moreno Maza
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Editor information

Editors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Technische Universität München, 85748, Garching, Germany

    Ernst W. Mayr

  4. Khristianovich Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Alvandi, P., Chen, C., Maza, M.M. (2013). Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2013. Lecture Notes in Computer Science, vol 8136. Springer, Cham. https://doi.org/10.1007/978-3-319-02297-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-02297-0_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02296-3

  • Online ISBN: 978-3-319-02297-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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