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Article overview
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Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials | | Hoon Hong
; Brittany Riggs
; | | Date: |
1 Jan 2023 | | Abstract: | Let $f$ be a monic univariate polynomial with non-zero constant term. We say
that $f$ is emph{positive/} if $f(x)$ is positive over all $xgeq0$. If all
the coefficients of $f$ are non-negative, then $f$ is trivially positive. In
1888, 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 emph{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. | | Source: | arXiv, 2301.00331 | | Services: | Forum | Review | PDF | Favorites | |
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