A New Proof of Meissner's Optimal Bound on the Degree of a Poincaré Multiplier and an Improved Optimal Degree Multiplier

Let $f$ be a monic univariate polynomial. We say that $f$ is positive if $f(x)$ is positive over all $x>0$. 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. In 1911, Meissner provided such a bound for quadratic polynomials. In this paper, we provide a linear algebra proof of Meissner’s optimal bound and compare an improved optimal degree Poincaré multiplier to one provided by Meissner.