An algorithm for primary decomposition in polynomial rings over the integers

Gerhard Pfister; Afshan Sadiq; Stefan Steidel

Open Mathematics (2011)

  • Volume: 9, Issue: 4, page 897-904
  • ISSN: 2391-5455

Abstract

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We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals, resp. over finite fields, and the idea of Shimoyama-Yokoyama, resp. Eisenbud-Hunecke-Vasconcelos, to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.

How to cite

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Gerhard Pfister, Afshan Sadiq, and Stefan Steidel. "An algorithm for primary decomposition in polynomial rings over the integers." Open Mathematics 9.4 (2011): 897-904. <http://eudml.org/doc/269723>.

@article{GerhardPfister2011,
abstract = {We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals, resp. over finite fields, and the idea of Shimoyama-Yokoyama, resp. Eisenbud-Hunecke-Vasconcelos, to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.},
author = {Gerhard Pfister, Afshan Sadiq, Stefan Steidel},
journal = {Open Mathematics},
keywords = {Gröbner bases; Primary decomposition; Modular computation; Parallel computation; primary decomposition; modular computation; parallel computation},
language = {eng},
number = {4},
pages = {897-904},
title = {An algorithm for primary decomposition in polynomial rings over the integers},
url = {http://eudml.org/doc/269723},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Gerhard Pfister
AU - Afshan Sadiq
AU - Stefan Steidel
TI - An algorithm for primary decomposition in polynomial rings over the integers
JO - Open Mathematics
PY - 2011
VL - 9
IS - 4
SP - 897
EP - 904
AB - We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals, resp. over finite fields, and the idea of Shimoyama-Yokoyama, resp. Eisenbud-Hunecke-Vasconcelos, to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.
LA - eng
KW - Gröbner bases; Primary decomposition; Modular computation; Parallel computation; primary decomposition; modular computation; parallel computation
UR - http://eudml.org/doc/269723
ER -

References

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  1. [1] Adams W.W., Loustaunau P., An Introduction to Gröbner Bases, Grad. Stud. Math., 3, American Mathematical Society, Providence, 1994 Zbl0803.13015
  2. [2] Ayoub C.W., The decomposition theorem for ideals in polynomial rings over a domain, J. Algebra, 1982, 76(1), 99–110 http://dx.doi.org/10.1016/0021-8693(82)90239-3 
  3. [3] Boege W., Gebauer R., Kredel H., Some examples for solving systems of algebraic equations by calculating Groebner bases, J. Symbolic Comput, 1986, 2(1), 83–98 http://dx.doi.org/10.1016/S0747-7171(86)80014-1 Zbl0602.65032
  4. [4] Decker W., Greuel G.-M., Pfister G., Primary decomposition: algorithms and comparisons, In: Algorithmic Algebra and Number Theory, Heidelberg, 1997, Springer, Berlin, 1999, 187–220 Zbl0932.13019
  5. [5] Decker W., Greuel G.-M., Pfister G., Schönemann H., Sincular3-1-1 - A computer algebra system for polynomial computations, 2010, http://www.singular.uni-kl.de 
  6. [6] Eisenbud D., Huneke C., Vasconcelos W., Direct methods for primary decomposition, Invent. Math., 1992, 110(2), 207–235 http://dx.doi.org/10.1007/BF01231331 Zbl0770.13018
  7. [7] Eisenbud D., Sturmfels B., Binomial ideals, Duke Math. J., 1996, 84(1), 1–45 http://dx.doi.org/10.1215/S0012-7094-96-08401-X 
  8. [8] Gianni P., Trager B., Zacharias G., Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput, 1988, 6(2–3), 149–167 http://dx.doi.org/10.1016/S0747-7171(88)80040-3 Zbl0667.13008
  9. [9] Gräbe H.-G., The SymbolicData Project - Tools and Data for Testing Computer Algebra Software, 2010, http://www.symbolicdata.org 
  10. [10] Greuel G.-M., Pfister G., A Singular Introduction to Commutative Algebra, 2nd ed., Springer, Berlin, 2008 
  11. [11] Greuel C.-M., Seelisch F., Wienand O., The Gröbner basis of the ideal of vanishing polynomials, J. Symbolic Comput., 2011, 46(5), 561–570 http://dx.doi.org/10.1016/j.jsc.2010.10.006 Zbl1210.13028
  12. [12] Seidenberg A., Constructions in a polynomial ring over the ring of integers, Amer. J. Math., 1978, 100(4), 685–703 http://dx.doi.org/10.2307/2373905 Zbl0416.13013
  13. [13] Shimoyama T, Yokoyama K., Localization and primary decomposition of polynomial ideals, J. Symbolic Comput., 1996, 22(3), 247–277 http://dx.doi.org/10.1006/jsco.1996.0052 Zbl0874.13022
  14. [14] Wienand O., Algorithms for Symbolic Computation and their Applications, Ph.D. thesis, Kaiserslautern, 2011 

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