Abstract
The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC’2007 paper and could be viewed as an analogue of comprehensive Gröbner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the RegularChains library in Maple. Following our previous work on generic regular decomposition for parametric polynomial systems, we introduce in this paper a so-called hierarchical strategy for computing CTDs. Roughly speaking, for a given parametric system, the parametric space is divided into several sub-spaces of different dimensions and we compute CTDs over those sub-spaces one by one. So, it is possible that, for some benchmarks, it is difficult to compute CTDs in reasonable time while this strategy can obtain some “partial” solutions over some parametric sub-spaces. The program based on this strategy has been tested on a number of benchmarks from the literature. Experimental results on these benchmarks with comparison to RegularChains are reported and may be valuable for developing more efficient triangularization tools.
The work was supported by National Science Foundation of China (Grants 11290141 and 11271034).
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Alvandi, P., Chen, C., Moreno Maza, M.: Computing the limit points of the quasi-component of a regular chain in dimension one. Computer Algebra in Scientific Computing, 30–45 (2013)
Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comp. 28, 105–124 (1999)
Chen, C., Davenport, J., May, J.P., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. In: Proc. ISSAC, pp. 187–194 (2010)
Chen, C., Davenport, J., Moreno Maza, M., Xia, B., Xiao, R.: Computing with semi-algebraic sets represented by triangular decomposition. In: Proc. ISSAC, pp. 75–82 (2011)
Chen, C., Golubitsky, O., Lemaire, F., Maza, M.M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007)
Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comp. 47 (6), 610–642 (2012)
Chou, S.-C.: Mechanical geometry theorem proving. Springer (1988)
Chen, Z., Tang, X., Xia, B.: Generic regular decompositions for parametric polynomial systems. Accepted by Journal of Systems Science and Complexity (2013), arXiv:1301.3991v1
Gao, X.-S., Chou, S.-C.: Solving parametric algebraic systems. In: Proc. ISSAC, pp. 335–341 (1992)
Gao, X.-S., Hou, X., Tang, J., Chen, H.: Complete solution classification for the perspective-three-point problem. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(8), 930–943 (2003)
Kalkbrener, M.: A generalized euclidean algorithm for computing for computing triangular representationa of algebraic varieties. J. Symb. Comp. 15, 143–167 (1993)
Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive gröbner systems. In: Proc. ISSAC, pp. 25–28 (2010)
Moreno Maza, M.: On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd., Oxford, UK (1999)
Montes, A., Recio, T.: Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 113–138. Springer, Heidelberg (2007)
Nabeshima, K.: A speed-up of the algorithm for computing comprehensive gröbner systems. In: Proc. ISSAC, pp. 299–306 (2007)
Suzuki, A., Sato, Y.: An alternative approach to comprehensive gröbner bases. In: Proc. ISSAC, pp. 255–261 (2002)
Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive gröbner bases. In: Proc. ISSAC, pp. 326–331 (2006)
Tang, X., Chen, Z., Xia, B.: Generic regular decompositions for generic zero-dimensional systems. Accepted by Science China: Information Sciences (2012), doi: 10.1007/s11432-013-5057-5
Wang, D.K.: Zero decomposition algorithms for system of polynomial equations. In: Computer Mathematics, pp. 67–70. World Scientific (2000)
Wang, D.M.: Computing triangular systems and regular systems. J. Symb. Comp. 30, 221–236 (2000)
Wang, D.M.: Elimination methods. Springer (2001)
Wang, D.M.: Elimination practice: software yools and applications. Imperial College Press (2004)
Weispfenning, V.: Comprehensive gröbner bases. J. Symb. Comp. 14, 1–29 (1992)
Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometries. Science in China Series A Mathematics, 507–516 (1977) (in Chinese)
Yang, L., Hou, X., Xia, B.: A complete algorithm for automated discovering of a class of inequality-type theorems. Science in China Series F Information Sciences 44(1), 33–49 (2001)
Yang, L., Xia, B.: Automatic inequality proving and discovering. Science Press (2008) (in Chinese)
Yang, L., Zhang, J.: Searching dependency between algebraic equations: An algorithm applied to automated reasoning. In: International Centre for Theoretical Physics, pp. 1–12 (1990)
Yang, L., Zhang, J., Hou, X.: Non-linear algebraic formulae and theorem automated proving. Shanghai Education Technology Publishers (1992) (in Chinese)
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Chen, Z., Tang, X., Xia, B. (2014). Hierarchical Comprehensive Triangular Decomposition. In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_66
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