The Picard type theorem for essential singularities of solutions of systems of n rational differential equations (Q915934)

Let \({\mathcal O}_ D\) be the integral domain of holomorphic functions on the domain \(D\subset {\mathbb{C}}\), and let \({\mathcal O}_ D[y_ 1,...,y_ n]\) be the polynomial ring in \(y_ 1,...,y_ n\) over \({\mathcal O}_ D\). The author considers a system of n rational differential equations \[ (*)\quad dy_ i/dx=P_ i(x,y_ 1,...,y_ n)/Q_ i(x,y_ 1,...,y_ n),\quad i=1,...,n, \] where \(P_ i,Q_ i\in {\mathcal O}_ D[y_ 1,...,y_ n]\) are relatively prime. Under some assumptions, to complicated to be presented here, he studies cluster sets of solutions of (*) at essential singularities and proves the Picard type theorems. The obtained results extend the known theorems on essential singularities by T. Kimura.