Abstract
In this paper, we introduce a modified Van der Waerden algorithm to decompose a variety into the union of irreducible varieties. We give an effective representation for irreducible varieties obtained by the algorithm, which allows us to obtain an irredundant decomposition easily. We show that in the zero dimensional case, the polynomial systems for the irreducible varieties obtained in the Van der Waerden algorithm generate prime ideals. As a consequence, we have an algorithm to decompose the radical ideal generated by a finite set of polynomials as the intersection of prime ideals and the degree of the polynomials in the computation is bounded by O(d n) where d is the degree of the input polynomials and n is the number of variables.
Partially supported by a National Key Basic Research Project of China under Grant No. 2004CB318000.
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References
Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional Systems theory, D.Reidel Publ. Comp. (1985)
Canny, J.: Generalised characteristics polynomials. Journal of Symbolic Computation 9, 241–250 (1990)
Chistov, A.: Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time. J. Sov. Math. 4, 1838–1882 (1986)
Chou, S.C.: Mechanical geometry theorem proving. D.Reidel Publishing Company, Dordrecht (1988)
Chou, S.C., Gao, X.S.: Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 207–220. Springer, Heidelberg (1990)
Elkadi, M., Mourrain, B.: A new algorithm for the geometric decomposition of a variety. In: Proc. of ISSAC 1999, pp. 9–16. ACM Press, New York (1999)
Gao, X.S., Chou, S.C.: On the dimension for arbitrary ascending chains. Chinese Bull. of Scis. 38, 396–399 (1993)
Gao, X.S., Chou, S.C.: On the theory of resolvents and its applications. Systems Science and Mathematical Sciences 12, 17–30 (1999)
Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation 6, 149–167 (1988)
Laplagne, S.: An algorithm for the computation of the radical of an ideal. In: Proc. of ISSAC 2006, pp. 191–195. ACM Press, New York (2006)
Macaulay, F.S.: The algebraic theory of modular systems. Cambridge University Press, Cambridge (1916)
Sausse, A.: A new approach to primary decompositon. Journal of Symbolic Computation 31, 243–257 (2001)
Ritt, J.F.: Differential algebra, American Mathematical Society (1950)
Szántó, Á.: Computation with polynomial systems, PhD thesis, Cornell University (1999)
Van der Waerden, B.L.: Einfürung in die algebraischen geometrie. Springer, Berlin (1973)
Van der Waerden, B.L.: Modern algebra II. Frederick Ungar Pub., New York (1953)
Wang, D.M.: Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Computer Aided Geometric Design 9, 471–484 (1992)
Wang, D.M.: Decomposing algebraic varieties. In: Wang, D., Yang, L., Gao, X.-S. (eds.) ADG 1998. LNCS (LNAI), vol. 1669, pp. 180–206. Springer, Heidelberg (1999)
Wu, W.T.: Basic principles of mechanical theorem-proving in elementary geometries. Journal of System Science and Mathematical Sciences 4, 207–235 (1984); Re-published in Journal of Automated Reasoning 2, 221–252 (1986)
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Li, J., Gao, XS. (2008). A Modified Van der Waerden Algorithm to Decompose Algebraic Varieties and Zero-Dimensional Radical Ideals. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_21
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DOI: https://doi.org/10.1007/978-3-540-87827-8_21
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