Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials

TL;DR

This paper proves that the Curtiss bound on the Poincaré multiplier's degree, which depends only on the roots' angles, is optimal among all bounds based solely on these angles for positive univariate polynomials.

Contribution

The paper establishes the optimality of the Curtiss bound among all angle-dependent bounds for the degree of Poincaré multipliers.

Findings

Curtiss bound is optimal among angle-dependent bounds.

The bound is tight for polynomials of degree 1 or 2.

The result clarifies the limitations of angle-based bounds for higher degrees.

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, Poincar\'e 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 Poincar\'e 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.