Algebraic solutions of Jacobi equations

Coutinho, S.; Ferreira, Marcos

doi:10.1090/S0025-5718-09-02238-8

Algebraic solutions of Jacobi equations
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by S. C. Coutinho and Marcos da Silva Ferreira;

Math. Comp. 78 (2009), 2427-2433

DOI: https://doi.org/10.1090/S0025-5718-09-02238-8

Published electronically: May 1, 2009

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Abstract:

We propose an algorithm to compute exactly the algebraic solutions of Jacobi equations over the projective plane.

References

William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
Arjeh M. Cohen, Hans Cuypers, and Hans Sterk (eds.), Some tapas of computer algebra, Algorithms and Computation in Mathematics, vol. 4, Springer-Verlag, Berlin, 1999. MR 1679917, DOI 10.1007/978-3-662-03891-8
S. C. Coutinho, Indecomposable non-holonomic $\scr D$-modules in dimension 2, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 2, 341–355. MR 1998565, DOI 10.1017/S0013091501001018
S. C. Coutinho and L. Menasché Schechter, Algebraic solutions of holomorphic foliations: an algorithmic approach, J. Symbolic Comput. 41 (2006), no. 5, 603–618. MR 2209167, DOI 10.1016/j.jsc.2005.11.002
G. Darboux, Mémoire sur les équations différentielles algébriques du I$^{\mathrm {o}}$ ordre et du premier degré, Bull. des Sc. Math. (Mélanges) (1878), 60–96, 123–144, 151–200.
Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Springer-Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh, and UNIX). MR 1930604, DOI 10.1007/978-3-662-04963-1
G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 2.0.5. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 10757
C. Jacobi, De Integratione Aequationes Differentiallis $(A+A’x+A''y)(xdy-ydx)-(B+B’x+B''y)dy+(C+C’x+C''y)dx)=0$, J. für die reine und angewandte Mathematik (1842), pp. 1–4, and Ges. Werke, vol. 4, pp. 256–262.
Camille Jordan, Cours d’analyse de l’École polytechnique. Tome III, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1991 (French). Équations différentielles. [Differential equations]; Reprint of the third (1915) edition. MR 1188188
J. P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979 (French). MR 537038, DOI 10.1007/BFb0063393
Harrison Tsai and Uli Walther, Computing homomorphisms between holonomic $D$-modules, J. Symbolic Comput. 32 (2001), no. 6, 597–617. Effective methods in rings of differential operators. MR 1866706, DOI 10.1006/jsco.2001.0485