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

TL;DR

This paper investigates the Kovalevskaya exponents and initial condition spaces of quasi-homogeneous polynomial differential systems, providing new criteria for series convergence and an algorithm for constructing initial condition spaces, especially for Painlevé equations.

Contribution

It introduces a new convergence criterion for formal series solutions, improves the Painlevé test, and presents an algorithm to construct the space of initial conditions using weighted blow-ups based on Kovalevskaya exponents.

Findings

Derived a necessary and sufficient condition for series convergence.

Enhanced the Painlevé test for integrability.

Provided an explicit algorithm for constructing initial condition spaces.

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).