# Questions tagged [rouches-theorem]

For questions on Rouché's theorem, which relates the number of roots of two holomorphic functions $f,g$ in some bounded domain $D$ given that $|g|<|f|$ on $\partial D$.

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### Proving existence of a root in the unit disk using Rouché's Theorem

Let $a\in\mathbb{C}$, and let $n \ge 2$. Prove that the polynomial $2022+az+2023z^n$ has a root in the unit disk, $D(0,1)$. There's an algebraic way to solve this with Vieta's formulas, by observing ...
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### Missing detail FTA proof

In the answer: https://math.stackexchange.com/a/3741902/987127, I don't see at which point there is a strict inequality, to satisfy Rouche's theorem.
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### Find parameters $a,b,c \in\mathbb{R}$, such that following system is compatible

Given $$\begin{cases}2x-3y+4z-5t = 1 \\ x+9y+az+t=-3 \\ 5x-6y+10z-bt=c\end{cases}$$ how to find $a,b,c\in\mathbb{R}$ so that the system is compatible? How should one approach such a problem? Where to ...
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### Rouches proof question regarding the curve $\gamma$

https://en.wikipedia.org/wiki/Rouché%27s_theorem I got a question regarding the proof of Rouches theorem. The proof our prof gave us tells us that the contour $\partial K$ needs to be partwise smooth, ...
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### Rouché's theorem proof question

I'm currently studying for a re-exam in Complex analysis, and have a question regarding the proof for Rouches theorem. Below is the exact proof, word by word, our professor have presented. I ...
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### What is the number of zeros of the this polynomial? [closed]

I wish to know the number of zeros of the polynomial $z^{10}-6z^7+3z^3+1$ in $|z|<1$. Does it have something to do with Rouche's theorem?
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### Find how many solutions (counting multiplicity) the equation $\sin z = ez^4$ has on the unit disk $|z|<1$.

This is a problem from a past qualifying exam in complex analysis. I'm working through these to study for my own upcoming qual. For this question, I think my proof is fairly straightforward, but I'd ...
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### Find number of zeros of $z^{113}-180z^{23}+115z^{7}-32z^2-3z-10$ in annulus $1≤|z|≤2$ [closed]

I know I have to use Rouché's theorem but I can't figure out how to choose $f$ and $g$
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### Show $2z^5-6z^2+z+1$ has three zeroes in $\{z : 1<\vert z \vert < 2\}$

I need to show $$2z^5-6z^2+z+1$$ has three zeroes on $\{z : 1< \vert z \vert < 2\}.$ So do I split in into cases for when $\vert z \vert= 1$ and when $\vert z \vert = 2$ by Rouches theorem I can ...
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### Number of zeros inside unit disk

What is the number of zeros of the equation $ze^{3-z}-5^{2}=0$ inside the unit disk? I believe that that the answer is zero, by some kind of Rouche Theorem. But, I am not %100 sure about that. A ...
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### Show that $P(z)=a_kZ^k$ has $n$ zeros in $B(0,1)$.

Let $P(z)=\sum_{k=0}^n a_kz^k$ where $0＜a_1＜\cdots＜a_n$. Show that $P(z)$ has n zeros in unit disk $\Bbb D$. Moreover, show that $\sum_{k=0}^n a_kcoskt$ has 2n zeros in open interval $(0,2π)$. I want ...
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### Why is only the boundary considered when using Rouche's Theorem?

A typical setup for a Rouche's Theorem problem is one where you have a polynomial, say $f(z)$, defined on some domain $U$. For simplicity let's take $U$ to be the unit disk. What often happens is one ...
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### Proving the polynomial $P_t(z) = \sum_{j=0}^N a_j(t) z^j$ has exactly one zero if $P_0$ has a simple zero with $a_j(t)$ and $P_t(z)$ continuous in $t$
Let $P_t(z)$ be a polynomial in $z$ of degree $\leq N$ for each value of $t \in [0,1]$. Suppose that $P_t(z)$ depends continuously on $t$ in the sense that $$P_t(z) = \sum_{j=0}^N a_j(t) z^j$$ and ...