Are there non-holomorphic or non-analytic polynomials of several complex variables? 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.

 A: 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.
