It is known that the rings of complex entire functions and the rings of real analytic functions are actually gcd domains (see also this MSE post).

I just discovered (somewhat to my naive surprise) that for functions $f : \mathbb{R} \to \mathbb{R}$, being real analytic and real entire are not the same. In other words, having a local series expansion does not guarantee a global series expansion, a simple counterexample being provided by the function $f(x) = \frac{1}{1 + x^2}$. I am now wondering about the following

Question: Is the ring of real entire functions a gcd domain?

It seems that the usual proofs do not work as inverting a non-zero real entire function does not keep it real entire any more. Any help will be most appreciated.


I don't have enough reputation to comment, so I will answer. Since you say that the ring of complex entire functions is a gcd domain, can't you do the following? Take two real entire functions $f$ and $g$, complexify them to form complex entire functions, let $h = \text{gcd} (f, g)$ be written as $h = \alpha f + \beta g$, and then take the real part of the restriction of all of these functions on the real axis. That should give you a gcd.


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