Abstract
A new stability condition of monomial bases is introduced. This stability condition is stronger than Kapur-Sun-Wang’s one. Moreover, a new algorithm for computing comprehensive Gröbner systems, is also introduced by using the new stability condition. A number of segments generated by the new algorithm is smaller than that of segments of in Kapur-Sun-Wang’s algorithm.
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References
Becker, T.: On Gröbner bases under specialization. Applicable Algebra in Engineering, Communication and Computing 5, 1–8 (1994)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-3, A computer algebra system for polynomial computations (2011)
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)
Fortuna, E., Gianni, P., Trager, B.: Degree reduction under specialization. Journal of Pure and Applied Algebra 164(1), 153–163 (2001)
Gianni, P.: Properties of Gröbner bases under specializations. In: Davenport, J. (ed.) EUROCAL 1987, pp. 293–297. ACM Press (1987)
Kalkbrener, M.: Solving Systems of Algebraic Equations Using Gröbner Base. In: Davenport, J. (ed.) EUROCAL 1987. LNCS, vol. 378, pp. 293–297. Springer, Heidelberg (1987)
Kalkbrener, M.: On the stability of Gröbner bases under specializations. Journal of Symbolic Computation 24, 51–58 (1997)
Kapur, D.: An approach for solving systems of parametric polynomial equations. In: Saraswat, V., Hentenryck, P. (eds.) Principles and Practice of Constraint Programming, pp. 217–244. MIT Press (1995)
Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive Gröbner systems. In: Watt, S. (ed.) International Symposium on Symbolic and Algebraic Computation, pp. 29–36. ACM Press (2010)
Montes, A.: A new algorithm for discussing Gröbner basis with parameters. Journal of Symbolic Computation 33(1-2), 183–208 (2002)
Montes, A., Wibmer, M.: Gröbner bases for polynomial systems with parameters. Journal of Symbolic Computation 45(12), 1391–1425 (2010)
Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Brown, C. (ed.) International Symposium on Symbolic and Algebraic Computation, pp. 299–306. ACM Press (2007)
Noro, M., Takeshima, T.: Risa/Asir- A computer algebra system. In: Wang, P. (ed.) International Symposium on Symbolic and Algebraic Computation, pp. 387–396. ACM Press (1992)
Suzuki, A., Sato, Y.: An alternative approach to comprehensive Gröbner bases. Journal of Symbolic Computation 36(3-4), 649–667 (2003)
Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Dumas, J.-G. (ed.) International Symposium on Symbolic and Algebraic Computation, pp. 326–331. ACM Press (2006)
Weispfenning, V.: Comprehensive Gröbner bases. Journal of Symbolic Computation 14(1), 1–29 (1992)
Weispfenning, V.: Canonical comprehensive Gröbner bases. Journal of Symbolic Computation 36(3-4), 669–683 (2003)
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Nabeshima, K. (2012). Stability Conditions of Monomial Bases and Comprehensive Gröbner Systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_21
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DOI: https://doi.org/10.1007/978-3-642-32973-9_21
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