no. 2
Weights, Kovalevskaya exponents and the Painlevé property
[Poids, exposants de Kovalevskaya et propriété de Painlevé]
Chiba, Hayato1
1 Advanced Institute for Materials Research Tohoku University Sendai, 980-8577 (Japan)
Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 811-848.

On étudie les degrés pondérés des fonctions hamiltoniennes quasi homogènes des équations de Painlevé. On classifie un t-uplet d’entiers positifs, appelé poids régulier, satisfaisant certaines conditions liées à la théorie de la singularité. A chaque équation polynomiale de Painlevé est associé un poids régulier. Inversement, pour les cas des dimensions 2 et 4, on montre qu’il existe une équation différentielle satisfaisant la propriété de Painlevé associée à chaque poids régulier. Les exposants de Kovalevskaya des systèmes hamiltoniens quasi homogènes sont également étudiés au moyen de poids réguliers, la théorie des singularités et la théorie des systèmes dynamiques. On montre qu’il existe une correspondance biunivoque entre les solutions de la série de Laurent et des variétés stables du champ vectoriel associé obtenues par l’éclatement du système. Pour des équations autonomes de Painlevé en dimension 4, la surface de niveau des fonctions hamiltoniennes peut être décomposée en une union disjointe de variétés stables.

Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlevé equations are investigated. A t-uple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. For each polynomial Painlevé equation a regular weight is associated. Conversely, for 2 and 4-dim cases, it is shown that there exists a differential equation satisfying the Painlevé property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical systems theory. It is shown that there is a one-to-one correspondence between Laurent series solutions and stable manifolds of the associated vector field obtained by the blow-up of the system. For 4-dim autonomous Painlevé equations, the level surface of Hamiltonian functions can be decomposed into a disjoint union of stable manifolds.

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DOI : 10.5802/aif.3591
Classification : 34M55
Keywords: Painlevé equation, quasihomogeneous vector field, regular weight, Kovalevskaya exponent
Mot clés : Équation de Painlevé, champ vectoriel quasi homogéne, poids normal, exposant de Kovalevskaya

Chiba, Hayato 1

1 Advanced Institute for Materials Research Tohoku University Sendai, 980-8577 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Chiba, Hayato. Weights, Kovalevskaya exponents and the Painlevé property. Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 811-848. doi : 10.5802/aif.3591. https://aif.centre-mersenne.org/articles/10.5802/aif.3591/

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