Arguments of zeros of highly log concave polynomials

TL;DR

This paper investigates the zeros of highly log-concave polynomials, establishing conditions on coefficient ratios that guarantee roots lie in specific sectors or half-planes, extending classical results like Kurtz's theorem.

Contribution

It introduces a new parameter 1 (beta) to characterize the argument distribution of polynomial roots, extending Kurtz's theorem to broader classes of polynomials.

Findings

Beta > 1.45 ensures roots are in the left half-plane.

Determines the exact form of the function 0(3) relating beta to root arguments.

Provides a version of Kurtz's theorem for polynomials with real coefficients.

Abstract

For a real polynomial $p = \sum_{i=0}^{n} c_ix^i$ with no negative coefficients and $n\geq 6$, let $\beta (p) = \inf_{i=1}^{n-1} c_i^2/c_{i+1}c_{i-1}$ (so $\beta (p) \geq 1$ entails that $p$ is log concave). If $\beta(p) > 1.45...$, then all roots of $p$ are in the left half plane, and moreover, there is a function $\beta_0 (\theta)$ (for $\pi/2 \leq \theta \leq \pi$) \st $\beta \geq \beta_0(\theta)$ entails all roots of $p$ have arguments in the sector $| \arg z| \geq \theta$ 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 $\beta \geq 4$ entails all roots are real). We also prove a version of Kurtz's theorem with real (not necessarily nonnegative) coefficients.