Consider two real-valued real analytic functions $f$ and $g$. I want to prove that there exists a greatest common divisor $d$, which is a real analytic function. By greatest common divisor, I mean the following:
- Common divisor: There exist real analytic functions $q_1, q_2$ such that $f = dq_1, g = dq_2$, and
- Greateast: If there is any other function $d_1$ that satisfies 1. above, then there exists a real analytic function $q_3$ such that $d = d_1q_3$.
I am guessing that a proof could be derived from the Taylor series expansion, but I am not sure how to proceed.