## 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].

### 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)