Ultrapolynomial

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

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