Ultrapolynomial

In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense.

Definition
Let $$d \in \mathbb{N}$$ and $$K$$ a field (typically $$\mathbb{R}$$ or $$\mathbb{C}$$) equipped with a norm (typically the absolute value). Then a function $$P: K^d \rightarrow K$$ of the form $$P(x) = \sum_{\alpha \in \mathbb{N}^d} c_\alpha x^\alpha$$ is called an ultrapolynomial of class $$\left\{ M_p \right\}$$, if the coefficients $$c_\alpha$$ satisfy $$\left| c_\alpha \right| \leq C L^{\left| \alpha \right|}/M_\alpha$$ for all $$\alpha \in \mathbb{N}^d$$, for some $$L>0$$ and $$C>0$$ (resp. for every $$L>0$$ and some $$C(L)>0$$).