Abstract
Based on the rational univariate representation of zero-dimensional polynomial systems, Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system, which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial system. This paper is devoted to giving an improvement for the rational representation. The idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Gröbner system of a parametric polynomial system to reduce the number of branches. The authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm (Tan-Zhang’s algorithm) with a minimal Dickson basis Gm of a Grobner basis for the ideal, where Gm is smaller in size than G. Based on this, the authors give an improved algorithm. Moreover, the proposed algorithm has been implemented on the computer algebra system Maple. Experimental data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.
Similar content being viewed by others
References
Wu W T, Basic Principles of Mechanical Theorem Proving in Geometries, Vol. I: Part of Elementary Geometries, Science Press, Beijing, 1984 (in Chinese).
Becker T and Weispfenning V, Gröbner Bases, Springer-Verlag, New York, 1993.
Cox D, Little J, and O’shea D, Using Algebra Geometry, 2nd Edition, Springer, New York, 2005.
Auzinger W and Stetter H J, An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations, Inter., Series of Number. Math., 1988, 86: 11–30.
Stetter H J, Matrix eigenproblem are at the heart of polynomial system solving, ACM SIGSAM Bull., 1996, 30(4): 27–36.
Kobayashi H, Fujise T, and Furukawa A, Solving systems of algebraic equations, Poceedings of the ISSAC88, Springer Lecture Notes in Computer Science, 1988, 358: 139–149.
Yokoyama K, Noro M, and Takeshima T, Solutions of systems of algebraic equations and linear maps on residue class rings, J. Symbolic Comput., 1992, 14: 399–417.
Lazard D, Resolution des systemes d’equations algebriques, Theor. Comp. Sci., 1981, 15(1): 77–110.
Lazard D, Grobner bases, Gaussian elimination and resolution of system of algebraic equations, Proc. EUROCAL’83, Springer Lecture Notes in Computer Science, 1983, 162: 146–156.
Lazard D, Solving zero-dimensional algebraic systems, J. Symbolic Comput., 1992, 13: 117–133.
Rouillier F, Solving zero-dimensional systems through the rational univariate representation, Appl. Algebra ENngrg. Comm. Comput., 1999, 9(5): 433–461.
Noro M and Yokoyama K, A modular method to compute the rational univariate representation of zero-dimensional ideals, J. Symbolic Comput., 1999, 28: 243–263.
Ouchi K and Keyser J, Rational univariate reduction via toric resultants, J. Symbolic Comput., 2008, 43(11): 811–844.
Zeng G X and Xiao S J, Computing the rational univariate representations for zero-dimensional systems by Wu s method, Sci. Sin. Math., 2010, 40(10): 999–1016.
Ma X D, Sun Y, and Wang D K, Computing polynomial univariate representations of zero-dimensional ideals by Grobner basis, Sci. China Math., 2012, 55(6): 1293–1302.
Tan C, The rational representation for solving polynomial systems, Ph.D. thesis, Jilin University, Changchun, 2009 (in Chinese).
Tan C and Zhang S G, Computation of the rational representation for solutions of high-dimensional systems, Comm. Math. Res., 2010, 26(2): 119–130.
Shang B X, Zhang S G, Tan C, et al., A simplified rational representation for positive-dimensional polynomial systems and SHEPWM equation solving, Journal of Systems Science & Complexity, 2017, 30(6): 1470–1482.
Schost E, Computing parametric geometric resolutions, Applicable Algebra in Engineering, Communication and Computing, 2003, 13(5): 349–393.
Safey El Din M, Yang Z H, and Zhi L H, On the complexity of computing real radicals of polynomial systems, Proceedings of the ISSAC 2018, New York, USA, 351–358.
Kapur D, Sun Y, and Wang D K, An efficient algorithm for computing a comprehensive Grobner system of a parametric polynomial systems, J. Symbolic Comput., 2013, 49: 27–44.
Kalkbrener M, On the stability of Grobner bases under specializations, J. Symbolic Comput., 1997, 24: 51–58.
Montes A, A new algorithm for discussing Grobner basis with parameters, J. Symbolic Comput., 2002, 33: 183–208.
Suzuki A and Sato Y, A simple algorithm to compute comprehensive Grobner bases using Grobner bases, Poceedings of the ISSAC2006, New York, 2006, 326–331.
Nabeshima K, A speed-up of the algorithm for computing comprehensive Grobner systems, Poceedings of the ISSAC 2007, New York, 2007, 299–306.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China under Grant No. 11801558, the Chinese Universities Scientific Funds under Grant No. 15059002 and the CAS Key Project QYZDJ-SSW-SYS022.
This paper was recommended for publication by Editor FENG Ruyong.
Rights and permissions
About this article
Cite this article
Xiao, F., Lu, D., Ma, X. et al. An Improvement of the Rational Representation for High-Dimensional Systems. J Syst Sci Complex 34, 2410–2427 (2021). https://doi.org/10.1007/s11424-020-9316-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-020-9316-4