论文标题
Kalantari绑定在多项式零上的融合的建设性证明
A constructive proof of the convergence of Kalantari's bound on polynomial zeros
论文作者
论文摘要
在他的2006年论文中，Jin证明了Kalantari对多项式零的界限，由$ M \ leq 2 $索引，分别称为$ l_m $和$ u_m $，成为$ M \ rightarrow \ rightarrow \ infty $。也就是说，给定$ n $多项式$ p（z）$没有消失的原点和错误容忍度$ε> 0 $，jin证明存在$ m $，因此$ \ frac {l_m} {ρ_{ρ_{min}}> 1-ε$ \ left |ρ\右| $。在本文中，我们得出了一个产生这样的$ M $的公式，从而建立了证明Jin的定理。实际上，我们证明了更强的定理，从某种意义上说，这种收敛是统一的，其速率仅取决于$ n $和其他一些参数。我们还提供了实验结果，该结果表明了某些$ d \ ll 2 $的最佳m（渐近）$ o \ left（\ frac {1} {ε^d} \ right）$。这些结果的证明将表明Jin的方法以$ o \ left（\ frac {n} {ε^d} \ right）$时间运行，从而使其有效地隔离高度的多项式零。
In his 2006 paper, Jin proves that Kalantari's bounds on polynomial zeros, indexed by $m \leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $m\rightarrow\infty$. 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 $\frac{L_m}{ρ_{min}} > 1-ε$, where $ρ_{min} := \min_{ρ:p(ρ) = 0} \left|ρ\right|$. 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) $O\left(\frac{1}{ε^d}\right)$ for some $d \ll 2$. A proof of these results would show that Jin's method runs in $O\left(\frac{n}{ε^d}\right)$ time, making it efficient for isolating polynomial zeros of high degree.
