A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions

Coutinho, S. C.

doi:10.5802/aif.2308

A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions
[Une preuve constructive de la densité des équations de Pfaff sans solutions algébriques]

Coutinho, S. C.^1,²

¹ Universidade Federal do Rio de Janeiro Departamento de Ciência da Computação Instituto de Matemática P.O. Box 68530, 21945-970 Rio de Janeiro, RJ (Brazil)
² Programa de Engenharia de Sistemas e Computação COPPE, UFRJ, PO Box 68511 21941-972, Rio de Janeiro, RJ (Brazil)

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1611-1621.

Résumé
Abstract

Nous présentons une preuve constructive du fait que l’ensemble des équations de Pfaff sans solutions algébriques sur le plan projectif complexe est dense dans l’ensemble de toutes les équations algébriques de Pfaff d’un degré donné.

We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree.

MR Zbl

DOI : 10.5802/aif.2308

Classification : 11R04, 37F75, 34M45, 32S65
Keywords: Pfaff equation, singularity, algebraic solution
Mot clés : équations de Pfaff, singularité, solution algébrique

Affiliations des auteurs :

Coutinho, S. C. ^{1, 2}

@article{AIF_2007__57_5_1611_0,
     author = {Coutinho, S. C.},
     title = {A constructive proof of the {Density} {of~Algebraic} {Pfaff} {Equations} without {Algebraic} {Solutions}},
     journal = {Annales de l'Institut Fourier},
     pages = {1611--1621},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     doi = {10.5802/aif.2308},
     zbl = {1130.34065},
     mrnumber = {2364144},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2308/}
}

TY  - JOUR
AU  - Coutinho, S. C.
TI  - A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 1611
EP  - 1621
VL  - 57
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.2308/
DO  - 10.5802/aif.2308
LA  - en
ID  - AIF_2007__57_5_1611_0
ER  -

%0 Journal Article
%A Coutinho, S. C.
%T A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions
%J Annales de l'Institut Fourier
%D 2007
%P 1611-1621
%V 57
%N 5
%I Association des Annales de l’institut Fourier
%U https://www.numdam.org/articles/10.5802/aif.2308/
%R 10.5802/aif.2308
%G en
%F AIF_2007__57_5_1611_0

Coutinho, S. C. A constructive proof of the Density of Algebraic Pfaff Equations without Algebraic Solutions. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1611-1621. doi : 10.5802/aif.2308. https://www.numdam.org/articles/10.5802/aif.2308/

Bibliographie
Cité par

[1] Brunella, M. Some remarks on indices of holomorphic vector fields, Publ. Mat., Volume 41 (1997) no. 2, pp. 527-544 | EuDML | MR | Zbl

[2] Cerveau, D.; Neto, A. L. Holomorphic foliations in $CP (2)$ having an invariant algebraic curve, Ann. Sc. de l’Institute Fourier, Volume 41 (1991), pp. 883-903 | DOI | EuDML | Numdam | Zbl

[3] Coutinho, S. C.; Pereira, J. V. On the density of algebraic foliations without algebraic invariant sets, J. Reine Angew. Math., Volume 594 (2006), pp. 117-135 | DOI | MR | Zbl

[4] Coutinho, S. C.; Schechter, L. M. Algebraic solutions of Holomorphic Foliations: an Algorithmic Approach, Journal of Symbolic Computation, Volume 41 (2006), pp. 603-618 | DOI | MR | Zbl

[5] Cox, D.; Little, J.; O’Shea, D. Using algebraic geometry, Undergraduate Texts in Mathematics, Springer, New York, 1998 | Zbl

[6] Daly, T. Axiom: the thirty year horizon, volume 1: tutorial, Lulu Press, 2005

[7] Darboux, G. Mémoire sur les équations différentielles algébriques du I $^{o}$ ordre et du premier degré, Bull. des Sc. Math. (Mélanges) (1878), pp. 60–96, 123–144, 151–200 | EuDML | JFM | Numdam

[8] Jouanolou, J. P. Equations de Pfaff algébriques, Lect. Notes in Math., 708, Springer-Verlag, Heidelberg, 1979 | MR | Zbl

[9] Lang, S. Algebra, Addison-Wesley, Reading, 1974 | MR | Zbl

[10] Maciejewski, A. J.; Ollagnier, J. M.; Nowicki, A.; Strelcyn, J. -M. Around Jouanolou non-integrability theorem, Indag. Mathem., Volume 11 (2000), pp. 239-254 | DOI | MR | Zbl

[11] Neto, A. L. Algebraic solutions of polynomial differential equations and foliations in dimension two, Holomorphic Dynamics (Lect. Notes in Math.), Volume 1345, New York-Heidelberg-Berlin (1988), pp. 192-232 | MR | Zbl

[12] Ollagnier, J. M.; Nowicki, A.; Strelcyn, J. -M. On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development, Bull. Sci. math., Volume 123 (1995), pp. 195-233 | MR | Zbl

[13] Prelle, M. J.; Singer, M. F. Elementary first integrals of differential equations, Trans. Amer. Math. Soc., Volume 279 (1983) no. 1, pp. 215-229 | DOI | MR | Zbl

Cité par Sources :