Abstract
A characteristic set theory for partial difference polynomial systems is proposed. We introduce the concept of coherent and regular ascending chains and prove that a partial difference ascending chain is the characteristic set of its saturation ideal if and only if it is coherent and regular. This gives a method to decide whether a polynomial belongs to the saturation ideal of an ascending chain. We introduce the concept of strongly irreducible ascending chains and prove that a partial difference ascending chain is the characteristic set of a reflexive prime ideal if and only if it is strongly irreducible. This gives a simple and precise representation for reflexive prime ideals.
Supported by a National Key Basic Research Project of China (2004CB318000).
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References
Aubry, P.: Ensembles Triangulaires de polynômes et Résolution de Systèmes Algébriques, Implantation en Axiom. Thèse de l’université Pierre et Marie Curie (1999)
Aubry, P., Lazard, D., Moreno Maza, M.: On the Theory of Triangular Sets. Journal of Symbolic Computation 28, 105–124 (1999)
Bentsen, I.: The Existence of Solutions of Abstract Partial Difference Polynomial. Trans. of AMS 158, 373–397 (1971)
Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the Radical of a Finitely Generated Differential Ideal. In: Proc. of ISSAC 1995, pp. 158–166. ACM Press, New York (1995)
Boulier, F., Lemaire, F., Moreno Maza, M.: Well Known Theorems on Triangular Systems and the D5 Principle. In: Proc. of Transgressive Computing 2006, pp. 79–91 (2006)
Bouziane, D., Kandri Rody, A., Mârouf, H.: Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal. Journal of Symbolic Computation 31, 631–649 (2001)
Cheng, J.S., Gao, X.S., Yap, C.K.: Complete Numerical Isolation of Real Zeros in Zero-dimensional Triangular Systems. In: Prof. ISSAC 2007, pp. 92–99. ACM Press, New York (2007)
Chou, S.C.: Mechanical Geometry Theorem Proving. Kluwer Academic Publishers, Norwell (1987)
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)
Cohn, R.M.: Difference Algebra. Interscience Publishers (1965)
Dahan, X., Moreno Maza, M., Schost, E., Wu, W., Xie, Y.: Lifting Techniques for Triangular Decompositions. In: Proc. ISSAC 2005, pp. 108–115. ACM Press, New York (2005)
Gao, X.S., Chou, S.C.: A Zero Structure Theorem for Differential Parametric Systems. Journal of Symbolic Computation 16, 585–595 (1994)
Gao, X.S., Luo, Y.: A Characteristic Set Method for Difference Polynomial Systems. In: International Conference on Polynomial System Solving, November 24-26 (2004); Submitted to JSC
Gao, X.S., Luo, Y., Zhang, G.: A Characteristic Set Method For Ordinary Difference Polynomial Systems. MM-Preprints 25, 84–102 (2006)
Gao, X.S., van der Hoeven, J., Yuan, C., Zhang, G.: A Characteristic Set Method for Differential-Difference Polynomial Systems. In: MEGA 2007, Strobl, Austria (July 2007)
Hubert, E.: Factorization-free Decomposition Algorithms in Differential Algebra. Journal Symbolic Computation 29, 641–662 (2000)
Kolchin, E.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)
Kondratieva, M.V., Levin, A.B., Mikhalev, A.V., Pankratiev, E.V.: Differential and Difference Dimension Polynomials. Kluwer Academic Publishers, Dordrecht (1999)
Ritt, J.F.: Differential Algebra, Amer. Math. Soc. Colloquium (1950)
Ritt, J.F., Raudenbush Jr., H.W.: Ideal Theory and Algebraic Difference Equations. Trans. of AMS 46, 445–452 (1939)
Rosenfeld, A.: Specialization in Differential Algebra. Trans. Am. Math. Soc 90, 394–407 (1959)
Wang, D.: Elimination Methods. Springer, Berlin (2000)
Wu, W.T.: On the Decision Problem and the Mechanization of Theorem in Elementary Geometry. Scientia Sinica 21, 159–172 (1978)
Wu, W.T.: A Constructive Theorey of Differential Algebraic Geometry. Lect. Notes in Math, vol. 1255, pp. 173–189. Springer, Heidelberg (1987)
Wu, W.T.: Basic Principle of Mechanical Theorem Proving in Geometries (in Chinese). Science Press, Beijing (1984); English Edition. Springer, Wien (1994)
van der Hoeven, J.: Differential and Mixed Differential-Difference Equations from the Effective Viewpoint (preprints, 1996)
Yang, L., Zhang, J.Z., Hou, X.R.: Non-linear Algebraic Equations and Automated Theorem Proving (in Chinese). ShangHai Science and Education Pub., ShangHai (1996)
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Zhang, GL., Gao, XS. (2008). Properties of Ascending Chains for Partial Difference Polynomial Systems . 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_25
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DOI: https://doi.org/10.1007/978-3-540-87827-8_25
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