A domain $$\Theta _n$$ is called a Rolle’s domain if every complex polynomial p of degree n, satisfying $$p(i)=p(-i)$$ , has at least one critical point in it. In this paper, we find the smallest possible Rolle’s domain made up of two closed disks that are symmetric with respect to the real and the imaginary axes. This is a strengthening of the main result in Sendov and Sendov (Proc Am Math Soc 146(8):3367–3380, 2018).

中文翻译：

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复多项式罗尔定理的进一步强化

如果满足 $$p(i)=p(-i)$$ 的每个 n 阶复多项式 p 中至少有一个临界点，则域 $$\Theta _n$$ 被称为罗尔域。在本文中，我们找到了由两个关于实轴和虚轴对称的封闭圆盘组成的最小可能的罗尔域。这是对 Sendov 和 Sendov 主要结果的加强（Proc Am Math Soc 146(8):3367–3380, 2018）。