Definition 3

A function $f:\mathbf{R}^n\rightarrow \mathbf{C}$ has polynomial growth if there exists an integer $k\geqslant 1$ such that the mapping defined by $x\mapsto (1 +\|x\|)^{-k}f(x)$ is bounded on $\mathbf{R}^n$.

Every function with polynomial growth is locally bounded. Every polynomial function on $\mathbf{R}^n$ has polynomial growth.