Algebraic solutions of algebraic differential equations
The author gives a condition under which a second-order algebraic differential equation has an algebraic solution. Let \(a_0\dots, a_p\), \(b_0,\dots, q\) be nonzero entire functions of one variable such that they have a finite number of poles and without common zero, and consider the following equation: \[ (w'')^n= \Biggl(\sum^p_{i=0} a_i(z) w^i\Biggr)\Biggl/\Biggl(\sum^q_{i=0} b_i(z) w^i\Biggr). \] Suppose that the above equation has at least one nonconstant \(\nu\)-valued algebroid solution \(w\) on \(\mathbb{C}\). Then he proves that that \(w\) is algebraic if all \(a_i\), \(b_j\) are polynomials and \(p< n+ q\).
scientific article; zbMATH DE number 2165846