Generic regular decompositions for parametric polynomial systems. (English) Zbl 1327.93101

Summary: This paper presents a generalization of the authors’ earlier work. In this paper, the two concepts, Generic Regular Decomposition (GRD) and Regular-Decomposition-Unstable (RDU) variety introduced in the authors’ previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors’ previous work. Then, the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors’ previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.


93B11 System structure simplification
68W30 Symbolic computation and algebraic computation
Full Text: DOI arXiv


[1] Kapur, D; Sun, Y; Wang, D, A new algorithm for computing comprehensive Gröbner systems, 25-28, (2010)
[2] Montes, A; Recio, T; Botana, F (ed.); Recio, T (ed.), Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems, 113-138, (2007), Berlin Heidelberg · Zbl 1195.68093
[3] Nabeshima, K, A speed-up of the algorithm for computing comprehensive Gröbner systems, 299-306, (2007) · Zbl 1190.13025
[4] Suzuki, A; Sato, Y, An alternative approach to comprehensive Gröbner bases, 255-261, (2002) · Zbl 1072.68699
[5] Suzuki, A; Sato, Y, A simple algorithm to compute comprehensive Gröbner bases, 326-331, (2006) · Zbl 1356.13040
[6] Weispfenning, V, Comprehensive Gröbner bases, J. Symb. Comp., 14, 1-29, (1992) · Zbl 0784.13013
[7] Aubry, P; Lazard, D; Maza, M, On the theories of triangular sets, J. Symb. Comp., 28, 105-124, (1999) · Zbl 0943.12003
[8] Chen, C; Golubitsky, O; Lemaire, F; Maza, M; Pan, W, Comprehensive triangular decomposition, 73-101, (2007) · Zbl 1141.68677
[9] Gao, X; Chou, S, Solving parametric algebraic systems, 335-341, (1992) · Zbl 0925.13014
[10] Kalkbrener, M, A generalized Euclidean algorithm for computing for computing triangular representationa of algebraic varieties, J. Symb. Comput., 15, 143-167, (1993) · Zbl 0783.14039
[11] Maza, M, On triangular decompositions of algebraic varieties, (1999)
[12] Wang, D, Zero decomposition algorithms for system of polynomial equations, 67-70, (2000) · Zbl 0981.65063
[13] Wang, M, Computing triangular systems and regular systems, J. Symb. Comput., 30, 221-236, (2000) · Zbl 1007.65039
[14] Wu, W, Basic principles of mechanical theorem proving in elementary geometries, J. Syst. Sci. Math. Sci., 4, 207-235, (1984)
[15] Yang, L; Hou, X; Xia, B, A complete algorithm for automated discovering of a class of inequality-type theorems, Science in China, Series F, 44, 33-49, (2001) · Zbl 1125.68406
[16] Yang L and Xia B, Automated Proving and Discovering Inequalities, Science Press Beijing, 2008 (in Chinese). · Zbl 0947.03017
[17] Yang, L; Zhang, J, Searching dependency between algebraic equations: an algorithm applied to automated reasoning, 1-12, (1991)
[18] Wang M, Elimination Methods, Springer New York, 2001.
[19] Wang M, Elimination Practice: Software Tools and Applications, Imperial College Press London, 2004. · Zbl 1099.13047
[20] Tang, X; Chen, Z; Xia, B, Generic regular decompositions for generic zero-dimensional systems, Science China Information Sciences, 57, 1-14, (2014)
[21] Xia, B, DISCOVERER: A tool for solving semi-algebraic systems, ACM Commun. Comput. Algebra., 41, 102-103, (2007)
[22] Chou S, Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, 1987.
[23] Chen, C; Maza, M, Algorithms for computing triangular decomposition of polynomial systems, J. Symb. Comp., 47, 610-642, (2012) · Zbl 1264.12011
[24] Cox D, Little J, and O’Shea D, Using Algebraic Geometry, Springer, New York, 1998.
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