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：本文研究完全单调多项式（简称CMP）的插值问题，即系数为非负实数的多项式插值。具体而言，给定有限集合$S\subset \mathbb{R}_{>0} \times \mathbb{R}_{\geq 0}$，我们考虑由Berg[1985]引入的$S$的极小多项式，该多项式在“极小”意义下指其最终被所有其他插值$S$的CMP所优超。我们给出了当$S$的极小多项式存在时其次数的上界。此外，我们提出了一种利用符号序列序结构计算给定$S$的极小多项式的新算法。应用上述上界，我们还分析了包括本算法在内的极小多项式计算算法的计算复杂性。