New progress in theory of surfaces defined by ordinary differential equations (Q914906)

Starting point is the theorem: Each solution surface \(F(w,z)=C\) of the complex system \[ (E_ n)\quad dw/dT=W_ n(w,z),\quad dz/dT=Z_ n(w,z) \] with polynomials \(W_ n(w,z)\) and \(Z_ n(w,z)\) contains some singular points of the system \((E_ n)\). The importance of this theorem is shown in the paper. Assumed that the system \((E_ n)\) possesses \(n^ 2+n+1\) singular points with \(n^ 2\) finite singular points and \(n+1\) infinite singular points in the complex domain, there is proved a theorem on the canonical form of the global solution and given an estimate for the upper bound of \(N(n)\). The results are demonstrated by two examples.