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

Lin, Vladimir

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)

MSC:

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)

Keywords:

compositions of holomorphic mappings; algebraic functions; Teichmüller space