An Extension Result for Maps Admitting an Algebraic Addition Theorem

• Published in: The Journal of geometric analysis Vol. 29; no. 1; pp. 316 - 327
• Main Authors: Baro, E., de Vicente, J., Otero, M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 15-01-2019; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Addition theorem; Algebra; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Theorems; 14P20; Algebraic addition theorem; Rational addition theorem; Nash groups; 33E05; 32A20
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-018-9992-7

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