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Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply refer to every polynomial on $N$ complex variables? And if so, why the extra emphasis?

Note: I found a related question for one complex variable here and I suspect the answer to be the same, but I just wanted to make sure I don't miss anything.

EDIT: Example of use:

Lemma Let $K_1$, $K_2$ be two disjoint compact convex subsets of $\mathbb{C}^n$. Then any holomorphic function $h$ on an open set $V \supset K_1 \cup K_2$ can be uniformly approximated on $K_1 \cup K_2$ by analytic polynomials.

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posilon
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    One definition that I've seen is the following: a polynomial $p(x,y) \in \mathbb{C}[x,y]$ is said to be analytic if there exists a polynomial $f(z) \in \mathbb{C}[z]$ such that $f(x+iy) = p(x,y)$. – msteve Jul 15 '15 at 14:44
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    However, polynomials in many complex variables are indeed holomorphic (to see this, use the fact that a function in several complex variables is holomorphic if and only if it is holomorphic in each variable). – msteve Jul 15 '15 at 14:45
  • That's what I thought. But why the extra emphasis then? Is it common for papers in this area to consider polynomials in the respective $2N$ real variables? – posilon Jul 15 '15 at 14:50
  • I think that the author of the paper likely means something similar to the definition in my first comment, perhaps a version in more variables. If you edit your question to include the wording and context, perhaps I can say something more specific. – msteve Jul 15 '15 at 14:52
  • In real variable theory one usually distinguishes between "real-analytic" and "complex-analytic". I suppose that here that's the same. The polynomial $p(x, y)=x^2+y^2$, for example, can be seen as a function defined on the complex plane, but it is not complex-analytic. – Giuseppe Negro Jul 15 '15 at 14:52
  • @msteve I added an example, thanks for the suggestion. – posilon Jul 15 '15 at 15:05

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It is common in several (and one) complex variables to consider ${\mathbb C}^n$ as ${\mathbb R}^{2n}$. Quite often we deal with polynomials which are not holomorphic (more often than not in fact), that is polynomials in $x$ and $y$, or better, polynomials in $z$ and $\bar{z}$. In fact, every polynomial in $x$ and $y$ can be written as a polynomial in $z$ and $\bar{z}$ using the identities $x = \frac{z+\bar{z}}{2}$ and $y = \frac{z-\bar{z}}{2i}$. So often when people write a polynomial they will write $P(z,\bar{z})$ and then "holomorphic polynomial" means that $P$ contains no monomials that include $\bar{z}$. Actually this is a way of checking that a polynomial is a holomorphic function: expand in terms of $z$ and $\bar{z}$ and see if there are any terms with $\bar{z}$ in the result. That is then usually written as $P(z)$. All the statements above in higher dimensions are of course in terms of vectors $z$, $\bar{z}$, $x$ and $y$, the idea is the same as in one variable.

For example the boundary of the unit ball in ${\mathbb C}^2$ is given by the polynomial equation $|z_1|^2+|z_2|^2=1$, which is of course not a holomorphic polynomial. Given that nonholomorphic polynomials come up quite often in several complex variables, the general assumption then is that a "polynomial" is an expression in $z$ and $\bar{z}$, and one tends to emphasize when a polynomial is, in fact, holomorphic.

It is the same thing for functions in general. When one says "function" one just means a complex valued function defined on some domain and one has to say "holomorphic function" if that is what is meant, as in your example.

Jiri Lebl
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