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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 316–327 | Cite as

An Extension Result for Maps Admitting an Algebraic Addition Theorem

  • E. BaroEmail author
  • J. de Vicente
  • M. Otero
Article

Abstract

We prove that if an analytic map \(f:U\rightarrow \mathbb {C}^n\), where \(U\subset \mathbb {C}^n\) is an open neighborhood of the origin, admits an algebraic addition theorem, then there exists a meromorphic map \(g:\mathbb {C}^n\dasharrow \mathbb {C}^n\) admitting an algebraic addition theorem such that each coordinate function of f is algebraic over \(\mathbb {C}(g)\) on U (this was proved by Weierstrass for \(n=1\)). Furthermore, g admits a rational addition theorem.

Keywords

Algebraic addition theorem Rational addition theorem Nash groups 

Mathematics Subject Classification

32A20 33E05 14P20 

Notes

Acknowledgements

J. de Vicente thanks E. Pantelis for the support to attend “Summer School in Tame Geometry,” Konstanz, July 18–23, 2016, where the results of this paper were presented. The authors also would like to thank José F. Fernando for helpful suggestions on an earlier version of this paper, Mark Villarino for his comments, and the referee for letting us know about the Refs. [11] and [5]. All the authors were supported by Spanish MTM2014-55565 and MTM2017-82105-P. The second author was also supported by a Grant of the International Program of Excellence in Mathematics at Universidad Autónoma de Madrid.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Departamento de Álgebra, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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