Methods for "recognizing" a polynomial of several variables.

Question

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial.

1) Show that $f^{(n)}\equiv0$ for some $n\geq0$.

2) Show that $\displaystyle\lim_{z\to\infty}f(z)$ exists.

I suppose that the first one could be adopted to several variables, but I do not think the second one can be. Does anyone know other similar methods?

EDIT: I would also be interested in an answer to the same question with "rational function" subbed in for "polynomial".

Answer 1

You can use the fact that an entire function in several complex variables is a polynomial if and only if it is separately polynomial in each coordinate direction (i.e. when restricting to the coordinate complex lines with all coordinates but one fixed). (A similar fact also holds for rational functions.)

Then your condition 2 (or 1) can be restated to hold along each coordinate line.