Regular Chains under Linear Changes of Coordinates and Applications

  • Parisa AlvandiEmail author
  • Changbo Chen
  • Amir Hashemi
  • Marc Moreno Maza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)


Given a regular chain, we are interested in questions like computing the limit points of its quasi-component, or equivalently, computing the variety of its saturated ideal. We propose techniques relying on linear changes of coordinates and we consider strategies where these changes can be either generic or guided by the input.


Prime Ideal Limit Point Linear Change Zariski Closure Triangular Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvandi, P., Chen, C., Maza, M.M.: 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.) CASC 2013. LNCS, vol. 8136, pp. 30–45. Springer, Heidelberg (2013)Google Scholar
  2. 2.
    Aubry, P., Lazard, D., Maza, M.M.: On the theories of triangular Sets. J. Symb. Comput. 28(1–2), 105–124 (1999)Google Scholar
  3. 3.
    Boulier, F., Lemaire, F., Moreno Maza, M.: Pardi! In: Proceedings of International Symposium on Symbolic and Algebraic Computation, ISSAC 2001, pp. 38–47 (2001)Google Scholar
  4. 4.
    Boulier, F., Lemaire, F., Maza, M.M.: Computing differential characteristic sets by change of ordering. J. Symb. Comput. 45(1), 124–149 (2010)Google Scholar
  5. 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)Google Scholar
  6. 6.
    Chen, C., Maza, M.M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)Google Scholar
  7. 7.
    Dahan, X., Jin, X., Maza, M.M., Schost, É.: Change of order for regular chains in positive dimension. Theor. Comput. Sci. 392(1–3), 37–65 (2008)Google Scholar
  8. 8.
    Eisenbud, D.: Commutative Algebra with a View toward Algebraic Geometry. Springer, New York (1995)Google Scholar
  9. 9.
    Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2002)Google Scholar
  10. 10.
    Hashemi, A.: Effective computation of radical of ideals and its application to invariant theory. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 382–389. Springer, Heidelberg (2014)Google Scholar
  11. 11.
    Krick, T., Logar, A.: An algorithm for the computation of the radical of an ideal in the ring of polynomials. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC 1991. LNCS, vol. 539, pp. 195–205. Springer, Heidelberg (1991)Google Scholar
  12. 12.
    Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. of Complexity 19(4), 564–596 (2003)Google Scholar
  13. 13.
    Lemaire, F., Maza, M.M., Pan, W., Xie, Y.: When does 〈T〉 equal sat(T)? J. Symb. Comput. 46(12), 1291–1305 (2011)Google Scholar
  14. 14.
    Logar, A.: A computational proof of the noether normalization lemma. In: Mora, T. (ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC 1988. LNCS, vol. 357, pp. 259–273. Springer, Heidelberg (1989)Google Scholar
  15. 15.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Commun. Comput. 9(5), 433–461 (1999)Google Scholar
  16. 16.
    Seiler, W.M.: A combinatorial approach to involution and δ-regularity II: Structure analysis of polynomial modules with Pommaret bases. Appl. Alg. Eng. Comm. Comp. 20, 261–338 (2009)Google Scholar
  17. 17.
    Sommese, A.J., Verschelde, J.: Numerical homotopies to compute generic points on positive dimensional algebraic sets. J. Complexity 16(3), 572–602 (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Parisa Alvandi
    • 2
    Email author
  • Changbo Chen
    • 1
  • Amir Hashemi
    • 3
    • 4
  • Marc Moreno Maza
    • 2
  1. 1.Chongqing Key Laboratory of Automated Reasoning and CognitionChongqing Institute of Green and Intelligent Technology, Chinese Academy of SciencesBeijingChina
  2. 2.ORCCA, University of Western OntarioLondonCanada
  3. 3.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  4. 4.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations