Abstract
We introduce the notion of regular decomposition of an ideal and present a first algorithm to compute it. Designed to avoid generic perturbations and eliminations of variables, our algorithm seems to have a good behaviour with respect to the sparsity of the input system. Beside, the properties of the regular decompositions allow us to deduce new algorithms for the computation of the radical and the weak equidimensional decomposition of an ideal. A first implementation shows promising results.
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References
Bertini, E.: Sui sistemi lineari. Istit. Lombardo Accad. Sci. Lett. Rend. A Istituto 15(II), 24–28 (1982)
Eisenbud, D., Huneke, C., Vasconcelos, W.: Direct methods for primary decomposition. Invent. Math. 110(2), 207–235 (1992)
Wu, W.J.: On zeros of algebraic equations—an application of Ritt principle. Kexue Tongbao (English Ed. ) 31(1), 1–5 (1986)
Lazard, D.: A new method for solving algebraic systems of positive dimension. Discrete Appl. Math. 33(1-3), 147–160 (1991); Applied algebra, algebraic algorithms, and error-correcting codes, Toulouse (1989)
Kalkbrener, M.: Three Contributions to Elimination Theory. Technical report, Johannes Kepler University, Linz, Austria (1991)
Aubry, P., Moreno Maza, M.: Triangular sets for solving polynomial systems: a comparative implementation of four methods. J. Symbolic Comput. 28(1-2), 125–154 (1999); Polynomial elimination—algorithms and applications
Wang, D.: Computing triangular systems and regular systems. J. Symbolic Computation 30(2), 221–236 (2000)
On triangular decompositions of algebraic varieties, Technical Report TR 4/99, NAG Ltd., Oxford, UK. Presented at the MEGA 2000 (1999)
Lecerf, G.: Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions. In: Proceedings of ISSAC 2000. ACM, New York (2000)
Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. J. Symbolic Comput. 6(2-3), 149–167 (1988); Computational aspects of commutative algebra
Caboara, M., Conti, P., Traverso, C.: Yet another ideal decomposition algorithm. In: Mattson, H.F., Mora, T. (eds.) AAECC 1997. LNCS, vol. 1255, pp. 39–54. Springer, Heidelberg (1997)
Laplagne, S.: An algorithm for the computation of the radical of an ideal. In: ISSAC 2006, pp. 191–195. ACM, New York (2006)
Kalkbrener, M.: Algorithmic properties of polynomial rings. J. Symbolic Comput. 26(5), 525–581 (1998)
Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, Heidelberg (1994)
Matera, G., Turull Torres, J.M.: The space complexity of elimination theory: upper bounds. In: Foundations of computational mathematics, pp. 267–276. Springer, Heidelberg (1997)
Kaplansky, I.: Commutative rings. Revised edn. The University of Chicago Press, Chicago (1974)
Becker, T., Weispfenning, V.: Gröbner bases. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)
Decker, W., Greuel, G.M., Pfister, G.: Primary decomposition: algorithms and comparisons. In: Algorithmic algebra and number theory, pp. 187–220. Springer, Heidelberg (1997/1999)
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Moroz, G. (2008). Regular Decompositions. 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_22
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DOI: https://doi.org/10.1007/978-3-540-87827-8_22
Publisher Name: Springer, Berlin, Heidelberg
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