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Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials

Hoon Hong, Brittany Riggs

Abstract

Let $f$ be a monic univariate polynomial with non-zero constant term. We say that $f$ is positive if $f(x)$ is positive over all $x\geq0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that$f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. This naturally raised a challenge: find an upper bound on the smallest degree of multipliers. In 1918, Curtiss provided such a bound. Curtiss also showed that the bound is optimal (smallest) when degree of $f$ is 1 or 2. It is easy to show that the bound is not optimal when degree of $f$ is higher. The Curtiss bound is a simple expression that depends only on the angle (argument) of non-real roots of $f$. In this paper, we show that the Curtiss bound is optimal among all the bounds that depends only on the angles.

Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials

Abstract

Let be a monic univariate polynomial with non-zero constant term. We say that is positive if is positive over all . If all the coefficients of are non-negative, then is trivially positive. In 1883, Poincaré proved that is positive if and only if there exists a monic polynomial such that all the coefficients of are non-negative. Such polynomial is called a Poincaré multiplier for the positive polynomial . Of course one hopes to find a multiplier with smallest degree. This naturally raised a challenge: find an upper bound on the smallest degree of multipliers. In 1918, Curtiss provided such a bound. Curtiss also showed that the bound is optimal (smallest) when degree of is 1 or 2. It is easy to show that the bound is not optimal when degree of is higher. The Curtiss bound is a simple expression that depends only on the angle (argument) of non-real roots of . In this paper, we show that the Curtiss bound is optimal among all the bounds that depends only on the angles.
Paper Structure (8 sections, 13 theorems, 70 equations)

This paper contains 8 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Proof of Angle-Based Optimality (Theorem \ref{['thm:optimal']})
  4. Overview
  5. Proof of Angle-Based Optimality (Theorem \ref{['thm:optimal']})
  6. Common Notations
  7. Concerning Angles in Quadrant 1, $0<\theta<\dfrac{\pi}{2}$
  8. Concerning Angles in Quadrant 2, $\dfrac{\pi}{2}\leq\phi\leq\pi$

Key Result

Theorem 1

Let $r_{1}e^{\pm i\theta_{1}},\ldots,r_{\sigma}e^{\pm i\theta_{\sigma}}$ be the roots of $f$ where multiple roots are repeated. Let Then $opt\left( f\right) \leq b\left( f\right)$ and the equality holds if $\deg f\leq2$.

Theorems & Definitions (19)

  • Definition 1: Optimal Bound
  • Example 2
  • Theorem 1: Curtiss Bound 1918 C18
  • Example 3
  • Theorem 2: Main Result: Angle-Based Optimality of Curtiss's Bound
  • Example 4
  • Lemma 3: Coefficients
  • Lemma 4: All Angles in Quadrant 1
  • Lemma 5: One Less Degree
  • Lemma 6: 3 Coefficients
  • ...and 9 more