Movable singular points of polynomial ordinary differential equations (Q2387951)

The paper deals with movable singular points of polynomial ordinary differential equations of the form \[ w^{(n)}= P(w^{(n-1)}, w^{(n-2)},\dots, w,z),\tag{1} \] where \(n\) is a positive integer, \(z\) is the independent complex variable, \(w\) is a complex-valued function of \(z\), and \(P\) is a polynomial in \(w\) and its derivatives with coefficients analytic with respect to \(z\) in some domain \(U\) of the complex plane. The author shows that a nonlinear equation of the form (1) always admits movable singular points. If the equation (1) is rewritten in the form \[ w^{(n)}= \sum_{h\in S} a_h(z)(w^{(n-1)})^{h_{n-1}}\cdots (w')^{h_1} w^{h_0},\tag{2} \] where \(S\) is a set of \(n\)-tuples \(h =(h_0,h_1,\dots, h_{n-1})\) of nonnegative integers and the \(a_h\) are functions analytic in the domain \(U\). Let \(|h|= \sum^{n-1}_{i=0} h_i\), \(\nu(h)= \sum^{n-1}_{i=0} ih_i\), let \(\Theta= \max_{h\in S}\nu(h)\) be the maximum weight of terms on the right-hand side (2). Then, the author proves that for the nonlinear equation (2) to be free of movable singular points, it is necessary that \(\Theta< n\) and that the Bureau number of the equation is either 1 or 2.