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- W4313483669 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, 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 emph{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." @default.
- W4313483669 created "2023-01-06" @default.
- W4313483669 creator A5020004215 @default.
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- W4313483669 date "2022-12-31" @default.
- W4313483669 modified "2023-09-25" @default.
- W4313483669 title "Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials" @default.
- W4313483669 doi "https://doi.org/10.48550/arxiv.2301.00331" @default.
- W4313483669 hasPublicationYear "2022" @default.
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