A constructive proof of the convergence of Kalantari's bound on polynomial zeros

Abstract

In his 2006 paper, Jin proves that Kalantari's bounds on polynomial zeros, indexed by $m\le 2$ and called $L_m$ and $U_m$ respectively, become sharp as $m\to \infty$. That is, given a degree $n$ polynomial $p(z)$ not vanishing at the origin and an error tolerance $\epsilon >0$, Jin proves that there exists an $m$ such that $\frac{L_m}{{\rho }_{min}}>1-\epsilon$, where ${\rho }_{min}:={\min }_{\rho :p(\rho )=0}\left|\rho \right|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jin's theorem. In fact, we prove the stronger theorem that this convergence is uniform in a sense, its rate depending only on $n$ and a few other parameters. We also give experimental results that suggest an optimal m of (asymptotically) $O\left(\frac{1}{{\epsilon }^d}\right)$ for some $d\ll 2$. A proof of these results would show that Jin's method runs in $O\left(\frac{n}{{\epsilon }^d}\right)$ time, making it efficient for isolating polynomial zeros of high degree.

References in corpus (1)

• Collatz polynomials: an introduction with bounds on their zeros