### Abstract

Formal series solutions and the Kovalevskaya exponents of a quasi-homogeneous polynomial system of differential equations are studied by means of a weighted projective space and dynamical systems theory. A necessary and sufficient condition for the series solution to be a convergent Laurent series is given, which improves the well-known Painlevé test. In particular, if a given system has the Painlevé property, an algorithm to construct Okamoto's space of initial conditions is given. The space of initial conditions is obtained by weighted blow-ups of the weighted projective space, where the weights for the blow-ups are determined by the Kovalevskaya exponents. The results are applied to the first Painlevé hierarchy (2. m-th order first Painlevé equation).

Original language | English |
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Pages (from-to) | 7681-7716 |

Number of pages | 36 |

Journal | Journal of Differential Equations |

Volume | 259 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2015 Dec 15 |

Externally published | Yes |

### Keywords

- Kovalevskaya exponent
- Quasi-homogeneous vector field
- The first Painlevé hierarchy
- Weighted projective space

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics