# An extension result for maps admitting an algebraic addition theorem

## TL;DR

This paper extends Weierstrass's result from one dimension to multiple dimensions, showing that maps with an algebraic addition theorem can be extended to meromorphic maps with a rational addition theorem.

## Contribution

It generalizes Weierstrass's extension theorem for algebraic addition theorems from one-dimensional to multi-dimensional complex maps.

## Key findings

- Existence of a meromorphic extension with algebraic addition theorem
- Extension to higher dimensions beyond Weierstrass's original result
- The extended map admits a rational addition theorem

## Abstract

We prove that if an analytic map $f:=(f_1,\ldots ,f_n):U\subset \mathbb{C}^n\rightarrow \mathbb{C}^n$ admits an algebraic addition theorem then there exists a meromorphic map $g:=(g_1,\ldots ,g_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ admitting an algebraic addition theorem such that $f_1,\ldots ,f_n$ are algebraic over $\mathbb{C}(g_1,\ldots ,g_n)$ on $U$ (this was proved by K. Weierstrass in dimension $1$). Furthermore, $(g_1,\ldots ,g_n)$ admits a rational addition theorem.