Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk, $$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then $p^{\prime }$ has $n-1$ roots inside the unit disk and $m$ roots at distance at least $(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $n$ sufficiently large, each of the $m$ roots has a critical point at distance ${\sim}n^{-1}$.

中文翻译：

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高斯-卢卡斯定理的一个稳定版本和应用

让$p:\mathbb{C}\rightarrow \mathbb{C}$是多项式。高斯-卢卡斯定理指出它的临界点，$p^{\素数}(z)=0$, 都包含在其根的凸包中。我们证明了一个稳定性版本，其最简单的形式如下：假设$p$已$n+m$根，在哪里$n$在单位盘内，$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{和}~m~\text{在外面}~\min _{n+ 1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$然后$p^{\素数}$已$n-1$根在单位盘和$m$至少在远处扎根$(dn-m)/(n+m)>1$从原点和所涉及的常数是尖锐的。我们还讨论了一个配对结果：在上面的设置中，对于$n$足够大，每个$m$根在远处有一个临界点${\sim}n^{-1}$.