Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field

[摘要] 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 Painleve test. In particular, if a given system has the Painleve 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 Painleve hierarchy (2m-th order first Painleve equation). (C) 2015 Elsevier Inc. All rights reserved.

[发布日期] 2015-12-15

[关键词] Quasi-homogeneous vector field;Weighted projective space;Kovalevskaya exponent;The first Painleve hierarchy