Around the 13th Hilbert problem for algebraic functions. (English) Zbl 0846.32014

Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 307-327 (1996).
Let \(u_n (z_1, \dots, z_n)\) be the “universal \(n\)-valued entire algebraic function of \(n\) complex variables” defined by the \(n\)-th degree equation: \(u^n + z_1 u^{n - 1} + \cdots + z_n = 0\). In the context of algebraic functions, a query of the type of Hilbert’s 13th problem may be formulated as asking whether \(u_n\) can be expressed as a finite composition of algebraic functions of some fewer number of variables (univalued holomorphic functions of any number of variables being allowed gratis). The author defines a “restricted composition problem” by paying particular attention to the branch loci of the various functions involved. For this problem he surveys and proves some results. Some well-known facts about the nonexistence of global cross sections for the universal curve over the Teichmüller space of the sphere with \(k\) punctures are utilized.
For the entire collection see [Zbl 0828.00035].
Reviewer: S.Nag (Madras)


32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14H05 Algebraic functions and function fields in algebraic geometry
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)