The classical Gauss-Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows. We show that if the coefficients of a monic polynomial p(z) are in the sector {teiψ : ψ ∈ [0, φ], t ≥ 0}, for some φ ∈ [0, π), and the zeros are not in its interior, then the critical points of p(z) are also not in the interior of that sector. In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

中文翻译：

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高斯-卢卡斯定理的扇区类比

经典的高斯-卢卡斯定理指出，复系数多项式的临界点在其零点的凸包中。这个基本定理来自这样一个事实：如果多项式的所有零点都在一个半平面内，那么它的临界点也是如此。这项工作的主要结果是用扇形代替半平面，如下所示。我们证明，如果单数多项式 p(z) 的系数在扇区 {teiψ : ψ ∈ [0, φ], t ≥ 0} 中，对于某些 φ ∈ [0, π)，并且零点不在它的内部，那么 p(z) 的临界点也不在该扇区的内部。此外，我们给出了多项式满足主结果前提的必要条件。