# Methods for "recognizing" a polynomial of several variables.

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".

• Aside: $(2)$ should be simply that the limit exists: if $f$ is a constant polynomial, the limit won't be $\infty$.
– user14972
Sep 5, 2014 at 17:45
• @Hurkyl Thanks, correction made. Sep 5, 2014 at 19:12

• Actaully, remarkably, condition 2 cannot be so restated. In an answer here :(math.stackexchange.com/questions/1468743/…), a paper (American Mathematical Monthly 114 (2007), David Armitage: "Entire functions that tend to zero on every line") is referenced with an entire function which approaches zero along every line. I suspect there might similarly be an entire non-polynomial function which approaches $\infty$ along every line. Feb 7, 2017 at 19:45