Notes on algebraic functions (Q1864320)

Summary: Consideration of the monodromy group of the hypergeometric equation \(z(1-z)w''+ [\gamma-(1+\alpha +\beta)z]w'-\alpha \beta w=0\), in the case of \(\alpha = 1/6\), \(\beta = 5/6\), \(\gamma = 7/6\), shows that the global hypergeometric function solution \(\mathbf{F}(1/6;5/6;7/6;z)\) is nonalgebraic although it has only algebraic singularities. Therefore, the proposition given in [\textit{N. G. Chebotarëv}, Theory of algebraic functions (Russian), OGIZ, Moscow (1948); \textit{V. V. Golubev}, Lectures on the analytic theory of differential equations (Russian), Gos. Izdat. Tekhn.-Teor. Lit., Moscow (1950; Zbl 0038.24201)] that a function is algebraic if it has only the algebraic singularities on the extended \(z\)-plane is not true. Through introduction of the concept of singular element criterion for deciding when a function is algebraic on the basis of properties of its singularities is given.