Abstract
This paper generalizes the method of Ngô and Winkler (2010, 2011) for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation (AODE) to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.
Similar content being viewed by others
References
Ritt J F, Differential Algebra, American Mathematical Society, Colloquium Publications, New York, 1950.
Hubert E, The general solution of an ordinary differential equation, Proceedings of ISSAC 1996, ACM Press, New York, 1996.
Feng R Y and Gao X S, A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs, Journal of Symbolic Computation, 2006, 41: 739–762.
Feng R Y and Gao X S, Rational general solutions of algebraic ordinary differential equations, Proceedings of ISSAC 2004, ACM Press, New York, 2004.
Sendra J R and Winkler F, Tracing index of rational curve parametrization, Computer Aided Geometric Design, 2001, 18: 771–795.
Ngô L X C and Winkler F, Rational general solutions of first order non-autonomous parametrizable ODEs, Journal of Symbolic Computation, 2010, 45(12): 1426–1441.
Ngô L X C and Winkler F, Rational general solutions of planar rational systems of autonomous ODEs, Journal of Symbolic Computation, 2011, 46(10): 1173–1186.
Carnicer M M, The Poincaré problem in the nondicritical case, Annals of Mathematics, 1994, 140(2): 289–294.
Artin M and Mumford D, Some elementary examples of unirational varieties which are not rational, Proceedings of London Mathematical Society, 1972, 25(3): 75–95.
Castelnuovo G, Sulle superficie di genere zero, Memoria Scelte, Zanichelli, 1939, 307–334.
Zariski O, On Castelnuovo’s criterion of rationality p a = P 2 = 0 of an algebraic surface, Illinois Journal of Mathematics, 1958, 2: 303–315.
Kolchin E R, Differential Algebra and Algebraic Groups, Academic Press, 1973.
Wang D, Elimination Methods, Springer, Wien New York, 2001.
Hubert E, Factorization-free decomposition algorithms in differential algebra, Journal of Symbolic Computation, 2000, 29: 641–662.
Lazard D, A new method for solving algebraic systems of positive dimension, Discrete Applied Mathematics, 1991, 33: 147–160.
Kalkbrener M, A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, Journal of Symbolic Computation, 1993, 15(2): 143–167.
Aubry P, Lazard D, and Moreno Maza M, On the theories of triangular sets, Journal of Symbolic Computation, 1999, 28(1–2): 105–124.
Abhyankar S S and Bajaj C L, Automatic parametrization of rational curves and surfaces III: Algebraic plane curves, Computer Aided Geometric Design, 1988, 5: 390–321.
Sendra J R, Winkler F, and Pérez-Díaz S, Rational Algebraic Curves: A Computer Algebra Approach, Springer, Berlin, 2008.
Abhyankar S S and Bajaj C L, Automatic parametrization of rational curves and surfaces IV: Algebraic space curves, ACM Transactions on Graphics, 1989, 8(4): 325–334.
Lemaire F, Moreno Maza M, Pan W, and Xie Y, When does 〈T〉 equal sat(T)? Proceedings of ISSAC 2008, ACM Press, New York, 2008.
Jouanolou J P, Equations de Pfaff Algébriques, Lectures Notes in Mathematics, Vol. 708, Springer-Verlag, New York/Berlin, 1979.
Llibre J and Zhang X, Rational first integrals in the Darboux theory of integrability in ℂn, Bulletin des Sciences Mathématiques, 2010, 134: 189–195.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Austrian Science Foundation (FWF) via the Doctoral Program “Computational Mathematics” under Grant No. W1214, Project DK11, the Project DIFFOP under Grant No. P20336-N18, the SKLSDE Open Fund SKLSDE-2011KF-02, the National Natural Science Foundation of China under Grant No. 61173032, the Natural Science Foundation of Beijing under Grant No. 1102026, and the China Scholarship Council.
This paper was recommended for publication by Editor LI Ziming.
Rights and permissions
About this article
Cite this article
Huang, Y., Ngô, L.X.C. & Winkler, F. Rational general solutions of higher order algebraic odes. J Syst Sci Complex 26, 261–280 (2013). https://doi.org/10.1007/s11424-013-1252-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-013-1252-0