---
type: "problem"
title: "Lucas's theorem and an application"
slug: "lucass-theorem-and-an-application"
author: "sequoia"
tags: ["convergence", "convex hull", "polynomials", "roots", "sequence"]
difficulty: 65
qualityStatus: "unreviewed"
listed: true
origin: "Unknown"
originChapter: ""
originPage: ""
originNote: ""
license: "CC BY-SA 4.0"
---

1) Let $P$ be a complex polynomial with only simple roots. Show that all the roots of $P'$ belong to the convex hull of the roots of $P$, meaning the set $$\left\{\sum_{i=1}^d \lambda_i z_i/ \forall i\in[\![1,d]\!] \lambda_i\geqslant0, \sum_{i=1}^d\lambda_i=1\right\}.$$
2) Let $d\in\mathbb{N}^*$ and $u_0,\dots,u_{d-1}$ be some complex numbers and $(u_n)_{n\in\mathbb{N}}$ be the complex sequence defined by those first terms and the relation $$ \forall n\in\mathbb{N}, u_{n+d}=\frac{u_{n+d-1}+\dots+u_{n+1}+u_n}{d}.$$ Show that $u$ converges to a complex $l$ that you shall specify.