In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense.
Definition Let d ∈ N {\displaystyle d\in \mathbb {N} } and K {\displaystyle K} a field (typically R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ) equipped with a norm (typically the absolute value). Then a function P : K d → K {\displaystyle P:K^{d}\rightarrow K} of the form P ( x ) = ∑ α ∈ N d c α x α {\displaystyle P(x)=\sum _{\alpha \in \mathbb {N} ^{d}}c_{\alpha }x^{\alpha }} is called an ultrapolynomial of class { M p } {\displaystyle \left\{M_{p}\right\}} , if the coefficients c α {\displaystyle c_{\alpha }} satisfy | c α | ≤ C L | α | / M α {\displaystyle \left|c_{\alpha }\right|\leq CL^{\left|\alpha \right|}/M_{\alpha }} for all α ∈ N d {\displaystyle \alpha \in \mathbb {N} ^{d}} , for some L > 0 {\displaystyle L>0} and C > 0 {\displaystyle C>0} (resp. for every L > 0 {\displaystyle L>0} and some C ( L ) > 0 {\displaystyle C(L)>0} ).
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