Algebraic Solutions to a Differential Equation

Abstract

This chapter considers how Fuchs’ problem: when are all solutions to a linear ordinary differential equation algebraic? was approached, and solved, in the 1870s and 1880s. First, Schwarz solved the problem for the hypergeometric equation. Then Fuchs solved it for the general second-order equation by reducing it to a problem in invariant theory and solving that problem by ad hoc means. Gordan later solved the invariant theory problem directly. But Fuchs’ solution was imperfect, and Klein simplified and corrected it by a mixture of geometric and group-theoretic techniques which established the central role played by the regular solids already highlighted by Schwarz. Simultaneously, Jordan showed how the problem could be solved by purely group-theoretic means, by reducing it to a search for all finite monodromy groups of 2 × 2 matrices with complex entries and determinant 1. He was also able to solve it for 3rd and 4th order equations, thus providing the first successful treatment of the higher order cases, and to prove a general finiteness theorem for the nth order case (Jordan’s finiteness theorem). Later Fuchs and Halphen were able to treat some of these cases invariant-theoretically.