Arguments of zeros of highly log concave polynomials

Abstract

For a real polynomial $p={\sum }_{i=0}^n{c}_i{x}^i$ with no negative coefficients and $n\ge 6$, let $\beta (p)={\inf }_{i=1}^{n-1}{c}_i^2/c_{i+1}{c}_{i-1}$ (so $\beta (p)\ge 1$ entails that $p$ is log concave). If $\beta (p)>\mathrm{1.45...}$, then all roots of $p$ are in the left half plane, and moreover, there is a function ${\beta }_0(\theta )$ (for $\pi /2\le \theta \le \pi$) \st $\beta \ge {\beta }_0(\theta )$ entails all roots of $p$ have arguments in the sector $|\arg z|\ge \theta$ with the smallest possible $\theta$; we determine exactly what this function (and its inverse) is (it turns out to be piecewise smooth, and quite tractible). This is a one-parameter extension of Kurtz's theorem (which asserts that $\beta \ge 4$ entails all roots are real). We also prove a version of Kurtz's theorem with real (not necessarily nonnegative) coefficients.