Abstract
We generalize the method of Ngô and Winkler (J Symbolic Comput 46:1173–1186, 2011) for finding rational general solutions of a plane rational differential system to the case of a trivariate rational differential system. We give necessary and sufficient conditions for the trivariate rational differential system to have a rational solution based on proper reparametrization of invariant algebraic space curves. In fact, the problem for computing a rational solution of the trivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation. We prove that the linear rational solvability of this autonomous differential equation does not depend on the choice of proper parametrizations of invariant algebraic space curves. In addition, two different rational solutions corresponding to the same invariant algebraic space curve are related by a shifting of the variable. We also present a criterion for a rational solution to be a rational general solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abhyankar S.S., Bajaj C.L.: Automatic parametrization of rational curves and surfaces IV: algebraic space curves. ACM Trans. Graph. 8(4), 325–334 (1989)
Aubry P., Lazard D., Moreno Maza M.: On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)
Carnicer M.M.: The Poincaré problem in the nondicritical case. Ann. Math. 140, 289–294 (1994)
Feng R.Y., Gao X.S.: A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs. J. Symb. Comput. 41, 739–762 (2006)
Feng, R.Y., Gao, X.S.: Rational general solutions of algebraic ordinary differential equations. In: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC’2004), pp. 155–162. ACM Press, New York (2004)
Hubert E.: Factorization-free decomposition algorithms in differential algebra. J. Symb. Comput. 29, 641–662 (2000)
Hubert, E.: The general solution of an ordinary differential equation. In: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation (ISSAC’1996), pp. 189–195. ACM Press, New York (1996)
Kolchin E.R.: Differential algebra and algebraic groups. Academic Press, New York (1973)
Kalkbrener M.: A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15, 143–167 (1993)
Lazard D.: A new method for solving algbebraic systems of positive dimension. Discret. Apll. Math. 33, 147–160 (1991)
Lemaire, F., Moreno Maza, M., Pan, W., Xie, Y.: When does \({\langle T \rangle}\) equal sat(T)? In: Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation (ISSAC’2008), pp. 207–214, ACM Press, New York (2008)
Ngô L.X.C., Winkler F.: Rational general solutions of first order non-autonomous parametrizable ODEs. J. Symb. Comput. 45, 1426–1441 (2010)
Ngô L.X.C., Winkler F.: Rational general solutions of planar rational systems of autonomous ODEs. J. Symb. Comput. 46, 1173–1186 (2011)
Ritt J.F.: Differential algebra. American Mathematical Society, Colloquium Publications, New York (1950)
Sendra J.R., Winkler F.: Parametrization of algebraic curves over optimal field extensions. J. Symb. Comput. 23, 191–207 (1997)
Sendra J.R., Winkler F.: Tracing index of rational curve parametrization. Comput. Aided Geom. Des. 18, 771–795 (2001)
Sendra J.R., Winkler F., Pérez-Díaz S.: Rational algebraic curves: a computer algebra approach. Springer, Berlin (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, Y., Ngô, L.X.C. & Winkler, F. Rational General Solutions of Trivariate Rational Differential Systems. Math.Comput.Sci. 6, 361–374 (2012). https://doi.org/10.1007/s11786-012-0131-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-012-0131-8