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There are two results in Complex Analysis that have a counterpart in Algebra:

-If we consider the ring of holomorphic functions in an open set $\mathcal H(U)$ with the usual sum and product, every finitely generated ideal is principal. In fact it is generated by any holomorphic function that vanishes exactly where the ideal $I$ and with the same multiplicitiy.

This is the same as in $\mathbb C[X]$ (although here all the ideals are finitely generated), where every ideal is characterized by the zeroes and the multiplicities.

-In several complex variables, a function which is holomorphic in $U\setminus\{p\}$ is holomorphic in $U$.

If we restricted to rational functions $\displaystyle \frac{p(z)}{q(z)}$, this would be a corrolary from the fact that $q(z)=0$ has codimension $1$. Hence it can't be a point.

My question is if it is a coincidence that in these two cases, holomorphic functions act somewhat similar to polynomials, or if it is an instance of a more general phenomenon?

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General phenomenon. See en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry . –  Qiaochu Yuan Jun 10 '11 at 23:26
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As Qiaochu observes, there is a general theory of algebraic and analytic geometry. Let me try to sketch the ideas (at least, as I understand them).

The basic objects of algebraic geometry are the algebraic varieties: these are the spaces (to be formal, locally ringed spaces) that locally look like the zero set of a collection of polynomial equations in some $\mathbb{C}^n$ (or more generally over an algebraically closed field). The corresponding objects in complex analytic geometry are the analytic varieties: these are the locally ringed spaces that are locally given by the vanishing locus of a finite number of holomorphic functions $f_1, \dots, f_n$ in a ball $U \subset \mathbb{C}^n$ (and with a suitable sheaf of rings). There is a functor ("analytification") from complex algebraic varieties to complex analytic varieties. This functor is not full (there are analytic varieties, even compact ones, that do not arise from algebraic varieties), nor is it faithful (there are holomorphic functions which are not algebraic).

However, there are various comparison theorems known as GAGA (as they were introduced in Serre's paper, described in the Wikipedia link). Essentially, the result is that for a proper variety over $\mathbb{C}$, coherent sheaves on the algebraic variety and coherent sheaves on the analytification are the same thing (more formally, there is an equivalence of categories), and the cohomology (in degree zero, this means global sections) is the same. In particular, it follows that globally defined analytic functions are all algebraic. It still bears repeating that this GAGA story depends on a fixed complex proper variety (which is already given to be algebraic).

I don't know whether the observation you made fits into the GAGA story. But at least one should expect some similarities between the constructions one can make in the algebraic and analytic categories (though the techniques one needs are often different in the two instances).

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Excellent and informative answer as usual from you Akhil! –  Amitesh Datta Jun 11 '11 at 2:41
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@Amitesh: Dear Amitesh, thanks, and I'm glad you found it helpful. Cheers, –  Akhil Mathew Jun 11 '11 at 3:03
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