  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.
