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It is a deep theorem due to Oka that the structure sheaf of holomorphic functions $\mathcal{O}_{\mathbb{C}^n}$ is coherent.

Is it (at least in principle) possible to program this result as an algorithm? As an algorithm, this would correspond to specifying an open set $U$ and some holomorphic sections $s_1, \dots, s_n$ on that set and for the program to return a finite list of (holomorphic) relations that these sections satisfy. Presumably we can't specify any holomorphic sections but we could perhaps restrict ourselves to ones that can be written in terms of polynomials and exponentials.

Are there known results related to this?

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