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

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 improve the well known Painlev\'{e} test. In particular, if a given system has the Painlev\'{e} 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 Painlev\'{e} hierarchy ($2m$-th order first Painlev\'{e} equation).