On algebraic solutions of algebraic Pfaff equations

• Autorzy: Henryk Żołądek
• Języki publikacji: EN

Abstrakty

EN

We give a new proof of Jouanolou's theorem about non-existence of algebraic solutions to the system $ẋ=z^{s}, ẏ=x^{s}, ż=y^{s}$. We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.

Kategorie tematyczne

• 34A05: Explicit solutions and reductions
• 34C05: Location of integral curves, singular points, limit cycles

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 114
• Numer: 2
• Strony: 117-126

Daty

wydano

1995

otrzymano

1994-05-12

Twórcy

autor

Henryk Żołądek

• Institute of Mathematics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland

Bibliografia

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• [2] M. Carnicer, The Poincaré problem in the dicritical case, Ann. of Math. 140 (1994), 289-294.
• [3] D. Cerveau and A. Lins-Neto, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble) 41 (4) (1991), 883-903.
• [4] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. (Mélanges) 1878, 60-96; 123-144; 151-200.
• [5] J. P. Jouanolou, Equations de Pfaff algébriques, Lecture Notes in Math. 708, Springer, 1979.
• [6] A. Lins-Neto, Algebraic solutions of polynomial differential equations and foliations in dimension two, in: Holomorphic Dynamics, Lecture Notes in Math. 1345, Springer, 1988, 193-232
• [7] J. Moulin-Ollagnier, A. Nowicki and J.-M. Strelcyn, On non-existence of constants of derivations; the proof of Jouanolou theorem and its development, preprint, 1992.
• [8] H. Żołądek, The solution of the center-focus problem, preprint, 1992.