ar X iv : m at h - ph / 0 30 30 16 v 1 6 M ar 2 00 3 Hypergeometric solutions of some algebraic equations ∗

We give the hypergeometric solutions of some algebraic equations including the general fifth degree equation. 1. It is well known that the general algebraic equation of degree n ≥ 5 cannot be solved by radicales [Ab 1826]. However, it may be solved if we use wider classes of functions. For example, the fifth degree equation may be solved in modular functions [He 1858], [Kr 1858], [Kl 1884]; the general algebraic equation may be solved in hyperelliptic theta constants [Um 1984]. Another approach consists in the consideration of the algebraic solutions of the differential equations and was used in the classical Schwarz’s paper [Sch 1873], where all algebraic solutions of the standard hypergeometric equation were found. It was intensively developed by Poincaré. The results of Schwarz were completed in the paper [BH 1989] by the classification of the algebraic generalized hypergeometric functions nFn−1. ∗ This paper aroused from the problem of normalization of the ground state wave function of the trigonometric n-particle Calogero-Sutherland system (see the review paper [OP 1983]). The relation of this problem with the results of the present note will be considered later. On leave of absence from Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia. Current e-mail address: [email protected]