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S. C. Coutinho (2007)
Annales de l’institut Fourier
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.
Fernando Marcos, Edgar Pereira (2010)
Mathematica Bohemica
Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.
Enriquez, Benjamin, Gavarini, Fabio (2006)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Steven B. Bank (1972)
Compositio Mathematica
Wu, Derchyi (2002)
Electronic Journal of Differential Equations (EJDE) [electronic only]
Laurent Stolovitch (2005)
Publications Mathématiques de l'IHÉS
Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...
Yamada, Yasuhiko (2009)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Charles Osgood (1973)
Acta Arithmetica
Jean-Pierre Ramis, Yasutaka Sibuya (1994)
Annales de l'institut Fourier
We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equationthe idea of our proof is to interpret a formal power series solution...
Juan J. Morales-Ruiz (2002)
Banach Center Publications
We obtain an algebraic interpretation by means of the Picard-Vessiot theory of a result by Ziglin about the self-intersection of complex separatrices of time-periodically perturbed one-degree of freedom complex analytical Hamiltonian systems.
Steven B. Bank (1982)
Compositio Mathematica
Steven Bank (1972)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Steven B. Bank (1994)
Czechoslovak Mathematical Journal
Wang, Shupei (1995)
Annales Academiae Scientiarum Fennicae. Series A I. Mathematica
Suzuki, Takao (2010)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Staněk, Svatoslav (1986)
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
W. B. Jurkat, H. J. Zwiesler (1988)
Annales scientifiques de l'École Normale Supérieure
Steven Bank (1970)
Compositio Mathematica
Henryk Żołądek (2011)
Banach Center Publications
The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.
Renato Spigler, Marco Vianello (2000)
Archivum Mathematicum
We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary -progressive paths. This approach can provide higher flexibility in practical applications of the method.
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