Science China Mathematics

, Volume 55, Issue 6, pp 1293–1302 | Cite as

Computing polynomial univariate representations of zero-dimensional ideals by Gröbner basis



Rational Univariate Representation (RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals. In this paper, we will present a new algorithm to compute Polynomial Univariate Representation (PUR) of zero-dimensional ideals. The new algorithm is based on some interesting properties of Gröbner basis. The new algorithm also provides a method for testing separating elements.


RUR PUR zero-dimensional ideals Gröbner basis 


12Y05 13P10 13P15 33F10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Becker T, Weispfenning V. Gröbner Basis: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, 1993MATHGoogle Scholar
  2. 2.
    Cheng J S, Gao X S, Guo L L. Root isolation of zero-dimensional polynomial systems with linear univariate representation. J Symb Comp, 2012, 47: 843–8585MATHCrossRefGoogle Scholar
  3. 3.
    Emiris I Z, Pan V Y. Improved algorithms for computing determinants and resultants. J Complexity, 2005, 21: 43–71MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Faugère J C, Gianni P, Lazard D, et al. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J Symb Comp, 1993, 16: 329–344MATHCrossRefGoogle Scholar
  5. 5.
    Lazard D. Ideal bases and primary decomposition: case of two variables. J Symb Comp, 1985, 1: 261–270MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Marinari M G, Möller H M, Mora T. Gröbner bases of ideals defined by functionals with an application to ideals of projective points. Appl Algebra Engrg Comm Comp, 1993, 4: 103–145MATHCrossRefGoogle Scholar
  7. 7.
    Mourrain B, Técourt J P, Teillaud M. On the computation of an arrangement of quadrics in 3D. Comp Geom, 2005, 30: 145–164MATHCrossRefGoogle Scholar
  8. 8.
    Noro M, Yokoyama K. A modular method to compute the rational univariate representation of zero-dimensional ideals. J Symb Comp, 1999, 28: 243–263MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ouchi K, Keyser J. Rational univariate reduction via toric resultants. J Symb Comp, 2008, 43: 811–844MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Rouillier F. Solving zero-dimensional systems through the rational univariate representation. Appl Algebra Engrg Comm Comp, 1999, 9: 33–461MathSciNetGoogle Scholar
  11. 11.
    Sun Y, Wang D K. An efficient algorithm for factoring polynomials over algebraic extension field. Arxiv:0907.2300v2, 2009Google Scholar
  12. 12.
    Tan C, Zhang S G. Separating element computation for the rational univariate representation with short coefficients in zero-dimensional algebraic varieties. J Jilin Univ Sci, 2009, 47: 174–178MathSciNetMATHGoogle Scholar
  13. 13.
    Xiao S J, Zeng G X. Algorithms for computing the global infimum and minimum of a polynomial function. Sci China Math, 2012, 55: 881–891CrossRefGoogle Scholar
  14. 14.
    Zeng G X, Xiao S J. Computing the rational univariate representations for zero-dimensional systems by Wu’s method (in Chinese). Sci Sin Math, 2010, 40: 999–1016Google Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.KLMM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations