Say $f_1,...,f_k$ are given holomorphic functions, on, say, $\mathbb{C}$ (I'm working in a noncompact Riemann surface but it doesn't matter, $\mathbb{C}$ will do). Let $f$ be a fixed holomorphic function such that the order of $f$ at any point $z \in \mathbb{C}$ is equal to the minimum of the orders of $f_1,..f_n$, in other words, $f_i/f$ is always holomorphic for any $i = 1, \dots, n$. Now let $g$ be any holomorphic function. I want to show that there exist holomorphic functions $g_1,...,g_n$ such that $g_1f_1 + \cdots + g_nf_n = gf$. That's all I want, and I would love a bit of guidance please.
This is part of a bigger problem I'm working on and it seems true but for some reason I'm not seeing it...? Is the argument simple?
Since the order of $f$ is the minimum of the $f_i$'s, you can write $$ f=f_iu $$ with $u$ a unit in $\mathcal{O}_z$ for some $i$. But then the ideal $(f)\subset(f_1,f_2,...,f_n)$. Hence for any holomorphic function $g$, there are $g_j$'s holomorphic so that near $z$ $$gf=g_1f_1+g_2f_2+...g_nf_n\;.$$