In the Riemann sphere 1 is not summe of holomorphics map vanishing on 0 and $\infty$

Question

I want to prove (if it's right) that in the Riemann sphere one can not write the constant function 1 as a summe of two holomorphics map, one vanishing in 0 and one vanishing in $\infty$.

Answer 1

The answer depends on the interpretation of "holomorphic maps".

If we interpret that as "holomorphic maps $\widehat{\mathbb{C}} \to \mathbb{C}$", then, since the sphere $\widehat{\mathbb{C}}$ is compact, all holomorphic maps are constant (the restriction to $\mathbb{C}$ is a bounded entire function, hence constant by Liouville's theorem), and thus if a holomorphic map vanishes at some point of the sphere, it is identically $0$, and the sum of two holomorphic maps defined on the sphere cannot be $1$ if both functions have a zero.

If we interpret it as "holomorphic maps $\widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$", then things are different. Then holomorphic maps are rational functions, and we can write $1$ as the sum of two rational functions, one of which vanishes at $\infty$ and the other at $0$:

$$1 = \frac{1}{1+z} + \frac{z}{1+z}$$

is one of many examples.