# Tagged Questions

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### hexic polynomial question

I am faced with a polynomial of the form $$ax^6+bx^3+cx+d=0,$$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
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### Level curves of a polynomial and the zeros of its higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
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### poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
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### Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)

Let $p$ be a polynomial of degree $n\geq2$ and has $n$ different roots $z_1,\dots,z_n$. Prove that $\oint\frac{\mathrm{d}z}{p(z)}=0$ where the closed path is large enough so that all roots are in the ...
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### Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
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### Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
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### The degree of a map between complex projective lines

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$. How can I: show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$? ...
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### Dirichlet problem on a disk with polynomial boundary values

Suppose that $\phi$ is a real valued harmonic function on the unit disc that is continuous up to the boundary such that $\phi$ agree with a real valued polynomial on the unit circle. Then $\phi$ ...
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### Inverse functions of certain conformal maps

Suppose I have a family of conformal maps given by $\omega=z+az^n$ where $|a|\leq1/n$ which maps the unit disk onto the so called epitrochoid. I am wondering about the inverse conformal map which ...
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### Injective polynomial on the unit disc

Let $P(z)=\sum_{k=0}^{n}{a_kz^k}$ be polynomial that is injective in the open unit disc. Show that $|a_n|\le |a_1|/n$. I know that if $P$ is injective function than $P$ is conformal map and therefore ...
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### Solving polynomial equation using known trigonometric identity

I recently did an exam paper in which the following question was asked: Prove $$\sin(5\theta)=16\sin^{5}(\theta)-20\sin^{3}(\theta)+5\sin(\theta)$$ Hence find all the solutions to: ...
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### Polynomial equal to analytic function

Let p(z) be a polynomial of degree n. Show $\exists R$ and analytic $f(z)$ such that $p(z)=f(z)^n$ for $|z|>R$
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### Show that $P_a(z)=0$ iff $z=N(a)$ for polynomial $P_a$

Let for $a_0=(a_0,a_1,...,a_n)\in\mathbb C^{n+1}$ the polynomial $P_{a_0}=\sum_{k=0}^na_kz^k$ and $z_0\in\mathbb C$ with $P_{a_0}(z_o)=0$ and $DP_{a_o}$ (the differential matrix) invertible. Show ...
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### Application of Rouché's (Rouche's) Theorem to a Polynomial

Here is my question: State Rouché's theorem. How many roots of the polynomial $p(z) = z^8 + 3 z^7 + 6 z^2 + 1$ are contained in the annulus {$1 < |z| < 2$}? The statement is fine. I then ...
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### Prove that $p(z) = 2z^5 + 6z - 1$ have roots (in two sets)

Prove that $p(z) = 2z^5 + 6z - 1$ have one real root in $(0,1)$ and four root in $\left\{ z \in \mathbb{C} : 1<|z|<2 \right\}$. I suppose that we should use Rouché's theorem but I have no ...
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### Prove that $\bar{P}_{\bar{z}}=P_z,\ (P_z,P_z)=(P_{\bar{z}},P_{\bar{z}})$ with $P_z=\dfrac{\partial{P}}{\partial{z}}$

I have a problem: For $P$ is a nonzero real valued homogeneous polynomial of degree $k$: $$P(z,\bar{z})=\sum_{j=1}^{k-1}a_jz^j\bar{z}^{k-j}$$ where $a_j \in \Bbb C,\ a_j=\bar{a}_{k-j}$. ...
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### How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
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### Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
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### Tthe inverse of a Mellin transform of a polynomial…

Let $\mathcal{M}$ be the symbol of the Mellin transform as define in http://en.wikipedia.org/wiki/Mellin_transform In a calculus, I finally end up with $$\mathcal{M^{-1}(f)}=\mathcal{P}$$ where ...
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### Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?

Let $f$ be a complex function with the property that $f^{(n)} = 0$. Does this imply that $f$ is a polynomial? If so, why? Upon thinking about this problem myself, I can easily observe that every ...
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### Number of zeros of $z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
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### $0 < a_0 \leq a_1 \leq \cdots \leq a_n$, show $p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0$ has no root in $|z| < 1$

Assume $0 < a_0 \leq a_1 \leq \cdots \leq a_n$, show that equation $$p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0$$ does not have root in $|z| < 1$. Here $z$ is ...
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### $f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
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### Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
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### Do two distinct level sets determine a non-constant complex polynomial?

Let $f$ and $g$ be non-constant complex polynomials in one variable. Let $a\neq b$ be complex numbers and suppose $f^{-1}(a)=g^{-1}(a)$ and $f^{-1}(b)=g^{-1}(b)$. Does this imply $f=g$? If we think ...
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### Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
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### Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...
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### Complex roots of the polynomial $bz^{2}+2az+b$ are on the unit circle

I want to show that the roots of the polynomial $bz^{2}+2az+b$ ($a,b$ are real) when $\left|\frac{a}{b}\right|\leq1$ (which is equivalent to the discriminant not being positive) are on the unit ...
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### Question regarding asymptotics of complex rational functions

Suppose $P\left(z\right)$ and $Q\left(z\right)$ are complex polynomials such that $\deg Q=m\geq l=\deg P$ and wlog suppose the lead coefficient in both polynomials is $1$. I want to show that ...
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### If $f$ is entire and $\exp(f(z))$ is a polynomial, then $f$ is constant.

In a recent question that was just deleted, @danielfischer gave at the end of his answer the following exercise: for entire $f$, $$e^{f(z)} \text{ is a polynomial} \iff f \text{ is constant}$$ I was ...
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### Roots of $z^n+a_1z^{n-1}+\ldots+a_n$ lie inside $|z|\leq 1$

Let $P(z)=z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_{n}$, and suppose $|a_1|+\cdots+|a_n|\leq 1$. Find the least $R>0$ for which all the roots of $P(z)$ always lie inside $|z|\leq R$. $P(z)=z^n-1$ has ...
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### Point on unit circle such that $|(z-a_1)\cdots(z-a_n)|=1$

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. Prove that there exists a point $b$ on the unit circle such that $|P(b)|=1$. My solution: $|P(0)|=1$, and $P$ ...
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### conditions for the existence of complex roots:

find the necessary conditions under which the following polynomial will have non-real roots: $P(x)=Ax^3+Bx^2+x-D$ where $A>0$ and $D>0$. well if it has a+ib and a-ib as conjugate root then the ...
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### Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
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### Special polynomials having atleast one root on the unit circle

I have the following problem: For each $w\in\mathbb{T},$ ($\mathbb{T}$ denotes the unit circle), consider the polynomial $P_{w,n}(z)=z^{n+1}+z^n-2w$ of degree $n+1,$ where $n\in\mathbb{N}.$ Does there ...
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### Prove that $f(x)=(e^{ix}-e^{iz_0})f_1(x)$ where $f_1(x)$ is also a trigonometric polynomial

Let $f(x)=\sum_n c_ne^{inx}$ be a trigonometric polynomial. It then makes sense to define $f$ on $\mathbb{C}$ by allowing $x$ in this formula to be any complex number. Suppose $f(z_0)=0$ for some ...
Let $P_n$ be a polynomial of degree $n$ with complex coefficients. Does for any $l>0$ and small $\varepsilon>0$ there exist $C=C(l,\varepsilon)>0$ and $q=q(l,\varepsilon)>1$ s.t. in the ...
For two sets $A,B$, let $d(A,B)=\sup_{x\in A}\inf_{y\in B}|x-y|+\sup_{y\in B}\inf_{x\in A}|x-y|$. Let $p(z)=a_nz^n+\ldots+a_0$, and let $\epsilon>0$. Show that there exists $\delta>0$ such ...