A constructive proof of the convergence of Kalantaris bound on polynomial zeros

In his 2006 paper, Jin proves that Kalantaris bounds on polynomial zeros, indexed by $m leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $mrightarrowinfty$. 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 $rho_{min} := min_{rho:p(rho) = 0} left|rhoright|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jins 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) $Oleft(frac{1}{epsilon^d}right)$ for some $d ll 2$. A proof of these results would show that Jins method runs in $Oleft(frac{n}{epsilon^d}right)$ time, making it efficient for isolating polynomial zeros of high degree.