Movable singular points of algebraic ordinary differential equations (Q2468719)

The author finds necessary conditions for the absence of movable (critical) singular points in a general solution of an \(n\)-th order ODE \(w^{(n)}=F(w^{(n-1)},w^{(n-2)},\dots,w,z)\) with \(F\) rational in \(w\) and its derivatives. Namely, making use of explicit changes of variables involving a parameter \(\alpha\) and letting \(\alpha=0\), the author finds reduced equations which seem more accessible for searching for (critical) singular points. Then the necessity of the obtained conditions follows from the classical Poincaré theorem. In spite of the assertion that the author's theorem still relies on the study of the simplified equations, he is able to extract a corollary which claims that the above ODE without derivatives \(w^{(n-1)}\) and \(w^{(n-2)}\) always admits movable critical points, while the ODE not depending on \(w^{(n-1)}\) always admits movable singular (perhaps, noncritical) points.