Abstract
In this paper, the characterizability of radical differential ideals is discussed and the characterizability criterion for them is proved. This criterion is based on the known algorithm for decomposing a radical differential ideal into characterizable components. The inclusion problem is also considered, in the light of which the algorithm for minimal characteristic decomposition of principal ideals is discussed. This algorithm is improved by using the proven characterizability criterion. Differential and algebraic properties of autoreduced sets are discussed, and the structure of differential chains and characteristic sets of radical differential ideals is revealed. The corresponding algorithms are constructed and discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Kolchin, E.R., Differential Algebra and Algebraic Groups, Academic, 1973.
Ritt, J.F., Differential Algebra, New York: AMS, 1950.
Sit, W.Y., The Ritt—Kolchin Theory for Differential Polynomials, Differential Algebra and Related Topics (Proc. of the Int. Workshop NJSU, 2000), Li, G., Keigher, W.P., Cassidy, P.J., and Sit, W.Y., Eds., 2002.
Boulier, F., Lazard, D., Ollivier, F., and Petitot, M., Representation for the Radical of a Finitely Generated Differential Ideal, ISSAC, 1995, pp. 158–166.
Hubert, E., Factorization-free Decomposition Algorithms in Differential Algebra, J. Symbolic Computation, 2000, vol. 29, pp. 641–662.
Hubert, E., Essential Components of an Algebraic Differential Equation, J. Symbolic Computation, 1999, vol. 28,nos. 4–5, pp. 657–680.
Bouziane, D., Kandri Rodi, A., and Maârouf, H., Unmixed-Dimensional Decomposition of a Finitely Generated Perfect Differential Ideal, J. Symbolic Computation, 2001, vol. 31, pp. 641–662.
Kondrat'eva, M.V., Examples of Computation of Generators of a Differential Ideal by Its Characteristic Set, Programmirovanie, 2002, no. 2.
Cox, D., Little, J., and O'Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 1998. Translated under the title Idealy, mnogoobraziya i algoritmy, Moscow: Mir, 2000.
Cluzeau, T. and Hubert, E., Resolvent Representation for Regular Differential Ideals, Applicable Algebra Eng. Commun. Computing, 2003, vol. 13,no. 5, pp. 395–425.
Shafarevich, I.R., Osnovy algebraicheskoi geometrii, tom 2: Skhemy. Kompleksnye mnogobraziya (Fundamentals of Algebraic Geometry, vol. 2: Schemes and Complex Manifolds), Moscow: Nauka, 1988, 2nd ed.
Aubry, P., Lazard, D., and Moreno Maza M., On the Theories of Triangular Sets, J. Symbolic Computation, 1999, vol. 28, pp. 105–124.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ovchinnikov, A.I. Characterizable Radical Differential Ideals and Some Properties of Characteristic Sets. Programming and Computer Software 30, 141–149 (2004). https://doi.org/10.1023/B:PACS.0000029578.07393.29
Issue Date:
DOI: https://doi.org/10.1023/B:PACS.0000029578.07393.29