# Algebraic Solutions to a Differential Equation

## Abstract

This chapter considers how Fuchs’s problem: when are all solutions to a linear ordinary differential equation algebraic? was approached, and solved, in the 1870’s and 1880’s. 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’s 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 n^{th} order case (Jordan’s finiteness theorem). Later Fuchs and Halphen were able to treat some of these cases invariant-theoretically.

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## Notes

- 3.Kronecker's example is discussed in Neuenschwander, [1977, 7] which is based on Casorati's notes of a conversation with Kronecker, 16 October 1864.Google Scholar
- 4.See Kiernan [1972], Wussing [1969], Purkert [1971, 1973].Google Scholar
- 11.Sylow presented his discovery that subgroups of order p
^{r}exist in a group of order n, whenever p is a prime and p divides n, in [1872]. Jordan was much impressed with this result, and wrote to Sylow about it; Sylow’s letters are preserved in the collection of Jordan’s correspondence at the Ecole Polytechnique (catalogue numbers I, 19–22). Sylow’s proofs were permutation theoretic, and the history of their reformulation in abstract terms is well traced in Waterhouse [1980], Jordan himself never published a proof of Sylow’s theorems.Google Scholar - 14.In a letter from Jordan to Klein, 11 October 1878, [N.S.U. Bibliothek zu Göttingen, Klein, cod Ms. F. Klein, 10, number 11] Jordan wrote: “Mon cher ami “Vous avez parfaitement raison. En énumérant les groupes linéaires à trois variables, j’ai laissé échapper celui d’ordre 168 que vous me signalez”. Jordan corrected his mistake, and went on to discuss the group of the modular equation at the prime 11 (PSL (2; 11), of order 660) which Klein had presumably raised in his letter to Jordan. Jordan observed that the group has an element, say A′, of order 11, and one, B′, of order 5. If it is to be a linear group in three variables, which he doubted, then, he wrote: “En supposant qu’il n’y ait que 3[?] variables, il faudra trouver une substitution C′ telle que A′, B′, C′ combinées entre elles ne donnent que 12.11.5 substitutions. I1 ne serait pas difficile sans doute de vérifier si la chose est possible, mais je suis trop occupé en ce moment pour pouvoir exécuter ce calcule.” Jordan went on to discuss the Hessian group of order 216, and then congratulated Klein on his good fortune in working with Gordan. “Je ne me sens pas aucune en état de le suivre sur le terrain des formes ternaires; mais j’ai longuement approfondi sa belle démonstration de 1’Endlichkeit der Gründformen des formes b inaires...” Jordan’s intuition about PSL (2; 11) was correct, it cannot be represented as a group in three variables; however, both he and Klein missed a group of order 360 which can be represented by ternary collineations. This is Valentiner’s group (Valentiner, [1889]) which is discussed in Wiman [1896], where it is shown to be abstractly isomorphic to A
_{6}, the group of even permutations of 6 objects, and, in Klein, [1905 = 1922, 481-502.].Google Scholar - 15.See Hermite’s letter to P.du Bois-Reymond, 3 September 1877, in Hermite [1916], referred to in Hawkins, [1975, 93n].Google Scholar
- 16.Jordan was somewhat imprecise. Poincaré showed [1881x = Oeuvres III, 95–97] that to each finite monodromy group in three or more variables there correspond infinitely many differential equations having rational coefficients and algebraic solutions.Google Scholar