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Comprehensive Gröbner Systems and Bases

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The Gröbner Cover

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 27))

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Abstract

From now on we will develop the study of parametric polynomial systems of equations. In this chapter, after a brief introduction describing the progress in area, we will center our attention on Comprehensive Gröbner Systems (CGS) and Basis (CGB). Computing a Comprehensive Gröbner System is a necessary previous step for obtaining the complete canonical discussion provided by the Gröbner Cover, which is described in Chap. 5, after the introduction of I-regular functions in Chap. 4 that are also required.

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References

  1. D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms, 3rd ed., Springer, New-York, 1992.

    Book  Google Scholar 

  2. A. Dolzmann, T. Sturm, Redlog: Computer algebra meets computer logic, ACM Sigsam Bulletin 31/2 (1997) 2–9.

    Article  Google Scholar 

  3. M. Kalkbrenner, On the stability of Gröbner bases under specializations, Journal of Symbolic Computation 24 (1997) 51–58.

    Article  MathSciNet  Google Scholar 

  4. D. Kapur, Principles and Practices of Constraints Programming, chap. An Approach for Solving Systems of Parametric Polynomial Equations, MIT Press, 1995, pp. 217–244.

    Google Scholar 

  5. D. Kapur, Y. Sun, D. Wang, A new algorithm for computing comprehensive Gröbner systems, in: Proceedings of ISSAC’2010, ACM Press, New York, 2010, pp. 29–36.

    MATH  Google Scholar 

  6. M. Manubens, A. Montes, Improving dispgb algorithm using the discriminant ideal, Journal of Symbolic Computation 41 (2006) 1245–1263.

    Article  MathSciNet  Google Scholar 

  7. M. Manubens, A. Montes, Minimal canonical comprehensive Gröbner systems, Journal of Symbolic Computation 44 (5) (2009) 463–478.

    Article  MathSciNet  Google Scholar 

  8. A. Montes, A new algorithm for discussing Gröbner bases with parameters, Journal of Symbolic Computation 22 (2002) 183–208.

    Article  MathSciNet  Google Scholar 

  9. A. Montes, H. Schönemann, Singular “grobcov.lib” library D.2.4, http://www.singular.uni-kl.de, Computer Algebra System for polynomial computations. Center for Computer Algebra, University of Kaiserslautern, free software under the GNU General Public Licence (2017).

  10. A. Montes, M. Wibmer, Gröbner bases for polynomial systems with parameters, Journal of Symbolic Computation 45 (2010) 1391–1425.

    Article  MathSciNet  Google Scholar 

  11. K. Nabeshima, A speed-up of the algorithm for computing comprehensive Gröbner systems, in: Proceedings of ISSAC’2007, ACM Press, New York, 2007, pp. 299–306.

    MATH  Google Scholar 

  12. M. Rychlik, Complexity and applications of parametric algorithms of computational algebraic geometry, in: R. de la Llave, L. Petzold, J. Lorenz (eds.), Dynamics of Algorithms, vol. 118 of IMA Volumes in Mathematics and its Applications, Springer, 2000, pp. 1–29.

    Google Scholar 

  13. Y. Sato, A. Suzuki, An alternative approach to comprehensive Gröbner bases, Journal of Symbolic Computation 36:3–4 (2003) 649–667.

    MathSciNet  MATH  Google Scholar 

  14. A. Suzuki, Y. Sato, A simple algorithm to compute comprehensive Gröbner bases, in: Proceedings of ISSAC’2006, ACM, New York, 2006, pp. 326–331.

    MATH  Google Scholar 

  15. A. Suzuki, Y. Sato, Implementation of CGS and CGB on Risa/Asir and other computer algebra systems using Suzuki-Sato algorithm, ACM Communications in Computer Algebra 41:3.

    Google Scholar 

  16. V. Weispfenning, Comprehensive Gröbner bases, Journal of Symbolic Computation 14 (1992) 1–29.

    Article  MathSciNet  Google Scholar 

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Montes, A. (2018). Comprehensive Gröbner Systems and Bases. In: The Gröbner Cover . Algorithms and Computation in Mathematics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-030-03904-2_3

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