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

In his 2006 paper, Jin proves that Kalantari's bounds on polynomial zeros, indexed by m ≤ 2 and called L_m and U_m respectively, become sharp as m→∞. That is, given a degree n polynomial p(z) not vanishing at the origin and an error tolerance ϵ > 0, Jin proves that there exists an m such that L_m/ρ_min > 1-ϵ, where ρ_min := min_ρ:p(ρ) = 0|ρ|. 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(1/ϵ^d) for some d ≪ 2. A proof of these results would show that Jin's method runs in O(n/ϵ^d) time, making it efficient for isolating polynomial zeros of high degree.

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