Geodesic
On the generalized Ritt problem as a~computational problem
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 4, pp. 109-120.

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The Ritt problem asks if there is an algorithm that decides whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing whether a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of this equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking. 
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O. D. Golubitsky; M. V. Kondrat'eva; A. I. Ovchinnikov. On the generalized Ritt problem as a~computational problem. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 4, pp. 109-120. http://geodesic.mathdoc.fr/item/FPM_2008_14_4_a6/

[1] Koks D., Littl Dzh., O'Shi D., Idealy, mnogoobraziya i algoritmy, Mir, M., 2000

[2] Boulier F., Lazard D., Ollivier F., Petitot M., “Representation for the radical of a finitely generated differential ideal”, ISSAC' 95, Proc. 1995 Int. Symp. Symbolic and Algebraic Computation, ACM Press, New York, 1995, 158–166 | Zbl

[3] Boulier F., Lemaire F., “Computing canonical representatives of regular differential ideals”, ISSAC' 00, Proc. 2000 Int. Symp. Symbolic and Algebraic Computation, ACM Press, New York, 2000, 38–47

[4] Cohn R. M., “Specializations of differential kernels and the Ritt problem”, J. Algebra, 61:1 (1979), 256–268 | DOI | MR | Zbl

[5] Cohn R. M., “Valuations and the Ritt problem”, J. Algebra, 101:1 (1986), 1–15 | DOI | MR | Zbl

[6] Eisenbud D., Huneke C., Vasconcelos V., “Direct methods for primary decomposition”, Invent. Math., 110 (1992), 207–235 | DOI | MR | Zbl

[7] Golubitsky O., “Gröbner fan and universal characteristic sets of prime differential ideals”, J. Symbolic Comput., 41:10 (2006), 1091–1104 | DOI | MR | Zbl

[8] Golubitsky O., Kondratieva M. V., Ovchinnikov A., “Algebraic transformation of differential characteristic decompositions from one ranking to another”, J. Symbolic Comput., 44:4 (2009), 333–357 | DOI | MR | Zbl

[9] Hubert E., “Factorization-free decomposition algorithms in differential algebra”, J. Symbolic Comput., 29:4–5 (2000), 641–662 | DOI | MR | Zbl

[10] Hubert E., “Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems”, Symbolic and Numerical Scientific Computation, Second Int. Conf., SNSC 2001 (Hagenberg, Austria, September 12–14, 2001), Lect. Notes Comput. Sci., 2630, ed. F. Winkler, Springer, Berlin, 2003, 1–39 | MR | Zbl

[11] Hubert E., “Notes on triangular sets and triangulation-decomposition algorithms. II. Differential systems”, Symbolic and Numerical Scientific Computation, Second Int. Conf., SNSC 2001 (Hagenberg, Austria, September 12–14, 2001), Lect. Notes Comput. Sci., 2630, ed. F. Winkler, Springer, Berlin, 2003, 40–87 | MR | Zbl

[12] Kolchin E. R., Differential Algebra and Algebraic Groups, Academic Press, 1973 | MR | Zbl

[13] Kondratieva M. V., Levin A. B., Mikhalev A. V., Pankratiev E. V., Differential and Difference Dimension Polynomials, Kluwer Academic, 1999 | MR | Zbl

[14] Levi H., “On the structure of differential polynomials and on their theory of ideals”, Trans. Amer. Math. Soc., 51 (1942), 532–568 | DOI | MR | Zbl

[15] Miller R. G., “Computable fields and Galois theory”, Notices Amer. Math. Soc., 55:7 (2008), 798–807 | MR | Zbl

[16] Miller R. G., “Computability and differential fields: A tutorial”, Proc. Second Int. Workshop on Differential Algebra and Related Topics, eds. L. Guo, W. Sit, 2008 (to appear)

[17] Rabin M. O., “Computable algebras, general theory and theory of computable fields”, Trans. Amer. Math. Soc., 95:2 (1960), 341–360 | DOI | MR | Zbl

[18] Ritt J. F., Differential Algebra, Amer. Math. Soc., New York, 1950 | MR | Zbl

[19] Sit W. Y., “The Ritt–Kolchin theory for differential polynomials”, Proc. Int. Workshop on Differential Algebra and Related Topics (Rutgers, The State University of New Jersey, New Brunswick, USA, November 2–3, 2000), ed. L. Guo, World Scientific, Singapore, 2002, 1–70 | MR | Zbl

[20] Winkler F., Polynomial Algorithms in Computer Algebra, Springer, 1996 | MR