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

Katsuyuki Bando, Eitetsu Ken, Hirotaka Onuki

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.

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

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 , we consider \textit{the minimal polynomial} of , introduced by Berg [1985], which is `minimal,' in the sense that it is eventually majorized by all the other CMPs interpolating . We give an upper bound of the degree of the minimal polynomial of when it exists. Furthermore, we give another algorithm for computing the minimal polynomial of given 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.
Paper Structure (7 sections, 29 theorems, 56 equations, 3 figures, 3 algorithms)

This paper contains 7 sections, 29 theorems, 56 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A degree upper bound for minimal polynomials
  4. An order structure of sign sequences
  5. An algorithm along $\preceq$
  6. Open Problems
  7. Acknowledgement

Key Result

Theorem 2.1

Let $f \in \mathbb{R}[X] \setminus\{0\}$ and $p$ be the number of positive roots of $f$ counting their multiplicities. Then $(\mathop{\mathrm{SC}}\nolimits(\mathop{\mathrm{Sign}}\nolimits(f))-p)$ is an even non-negative integer. In particular,

Figures (3)

  • Figure : Minimal Polynomial by Linear Programming
  • Figure : Increment
  • Figure : Find the minimal polynomial

Theorems & Definitions (69)

  • Example 1.1
  • Theorem 2.1
  • Theorem 2.2: AlbouyFu
  • Corollary 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • Example 2.8
  • ...and 59 more