On algebraic solutions of algebraic Pfaff equations

Henryk Żołądek

Studia Mathematica (1995)

  • Volume: 114, Issue: 2, page 117-126
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Żołądek, Henryk. "On algebraic solutions of algebraic Pfaff equations." Studia Mathematica 114.2 (1995): 117-126. <http://eudml.org/doc/216183>.

@article{Żołądek1995,
abstract = {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.},
author = {Żołądek, Henryk},
journal = {Studia Mathematica},
keywords = {algebraic solution; algebraic Pfaff equation; polynomial vector field},
language = {eng},
number = {2},
pages = {117-126},
title = {On algebraic solutions of algebraic Pfaff equations},
url = {http://eudml.org/doc/216183},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Żołądek, Henryk
TI - On algebraic solutions of algebraic Pfaff equations
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 2
SP - 117
EP - 126
AB - 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.
LA - eng
KW - algebraic solution; algebraic Pfaff equation; polynomial vector field
UR - http://eudml.org/doc/216183
ER -

References

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  1. [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Dynamical Systems of Second Order, Moscow, Nauka, 1966 (in Russian). 
  2. [2] M. Carnicer, The Poincaré problem in the dicritical case, Ann. of Math. 140 (1994), 289-294. Zbl0821.32026
  3. [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. Zbl0734.34007
  4. [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. Zbl10.0214.01
  5. [5] J. P. Jouanolou, Equations de Pfaff algébriques, Lecture Notes in Math. 708, Springer, 1979. 
  6. [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 Zbl0677.58036
  7. [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. Zbl0855.34010
  8. [8] H. Żołądek, The solution of the center-focus problem, preprint, 1992. 

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