In this paper, we consider interpolation by \textit{completely monotonous} polynomials (CMPs for short), that is, polynomials with non-negative real coefficients. In particular, given a finite set $S\subset \mathbb{R}_{>0} \times \mathbb{R}_{\geq 0}$, we consider \textit{the minimal polynomial} of $S$, introduced by Berg [1985], which is `minimal,' in the sense that it is eventually majorized by all the other CMPs interpolating $S$. We give an upper bound of the degree of the minimal polynomial of $S$ when it exists. Furthermore, we give another algorithm for computing the minimal polynomial of given $S$ which utilizes an order structure on sign sequences. Applying the upper bound above, we also analyze the computational complexity of algorithms for computing minimal polynomials including ours.

翻译：本文研究由\textit{完全单调多项式}（简称CMPs）实现的插值问题，即具有非负实系数的多项式。特别地，给定有限集$S\subset \mathbb{R}_{>0} \times \mathbb{R}_{\geq 0}$，我们考察由Berg [1985]提出的$S$的\textit{极小多项式}，该多项式在“极小性”意义上具有如下性质：所有其他插值$S$的CMP最终都将控制该多项式。我们给出了极小多项式存在时其次数的上界。此外，我们提出了另一种计算给定$S$的极小多项式的算法，该算法利用了符号序列上的序结构。通过应用上述上界，我们还分析了包括本算法在内的计算极小多项式的算法的计算复杂度。