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arXiv · 2402.00409

On the Complexity of Interpolation by Polynomials with Non-negative Real Coefficients

Katsuyuki Bando·Eitetsu Ken·Hirotaka Onuki
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Abstract

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.

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Katsuyuki Bando, Eitetsu Ken, Hirotaka Onuki. 2024-02-01. On the Complexity of Interpolation by Polynomials with Non-negative Real Coefficients. https://arxiv.org/abs/2402.00409

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