Painleve Property and Algebraic Integrability of Single Variable Ordinary Differential Equations with Dominants

Abstract

If the general solution of an integrable nonlinear system has a movable singularity, we can often expand it in the Laurent series with enough arbitrary constants around the singular point in the complex domain. The property that a single valued general solution exists around a movable pole singular point is called property. There are many examples whose first integrals are found out when their parameters are chosen so that they have this property.l)-4) How does the property around a singular point relate to an integrability? In this regard, there are Ziglin's studY;) for a monodoromymatrix of Hamilton systems and Yoshida's study6) for Kowalevski's exponent (resonance) of similarity invariant systems. In this work we consider n-th order single variable ordinary differential equations (Unx-A(u, Ux, ', U(n-l)x)=O, AE e[u, ', U(n-l)x]) which have dominants, and we prove by an algebraic method that the equations satisfy Painleve property's necessary condition if they have enough algebraic first integrals.