1
vote
1answer
38 views

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 ...
4
votes
0answers
92 views

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 ...
0
votes
3answers
68 views

poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
1
vote
2answers
116 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
0
votes
3answers
45 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
5
votes
1answer
79 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
0
votes
3answers
60 views

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 ...
0
votes
4answers
87 views

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 ...
0
votes
1answer
34 views

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 ...
1
vote
1answer
17 views

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 $\mathbb{C}P^1\to\mathbb{C}P^1$? ...
0
votes
1answer
73 views

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 $\phi$ ...
1
vote
1answer
20 views

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 ...
0
votes
1answer
31 views

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 ...
0
votes
2answers
51 views

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: ...
0
votes
1answer
39 views

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 $|z|>R$
2
votes
0answers
26 views

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 ...
0
votes
1answer
40 views

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 ...
2
votes
2answers
46 views

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 ...
2
votes
1answer
41 views

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 $a_j \in \Bbb C,\ a_j=\bar{a}_{k-j}$. ...
2
votes
1answer
40 views

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 ...
2
votes
0answers
20 views

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 ...
0
votes
0answers
45 views

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 ...
0
votes
2answers
54 views

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 ...
0
votes
1answer
54 views

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 ...
3
votes
1answer
101 views

$0 < a_0 \leq a_1 \leq \cdots \leq a_n$, 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 ...
4
votes
6answers
190 views

$f(x)=x^3+ax^2+bx+c$ 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 ...
2
votes
0answers
72 views

Question on the Fourier Transform, specifically concerning polynomials

Suppose $P$ is a polynomial of degree $\ge 2$ with distinct roots, none lying on the real axis. Calculate: $$\int_{- \infty}^{\infty}\frac{e^{-2 \pi i x \xi}}{P(x)}dx,\space \space \space\xi \in ...
6
votes
1answer
343 views

Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
0
votes
1answer
110 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
0
votes
0answers
41 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
6
votes
3answers
237 views

Factor $x^4+1$ over $\mathbb{R}$

Factor $x^4+1$ over $\mathbb{R}$ Well, I read this question first wrongly, because the reader is about complex analysis, I did it for $\mathbb{C}$ first. I got. $x^4+1=(x-e^{\pi i/4 })(x-e^{3 ...
2
votes
1answer
77 views

Complex polynomial $p$ implies $p(z)=z^n$ or there exist a $z'$ with $|z'|=1$ such that $|p(z')|>1$

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{0}$ be a polynomial of degree $n\geq1$. How can you prove that either $p(z)=z^{n}$ or there exists $z'$ with $|z'|=1$ such that $|p(z')|>1$. Maybe we can use ...
1
vote
1answer
29 views

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$
3
votes
1answer
98 views

Minimize norm of a polynomial on a circle

Let $P=\sum_{k=0}^n a_kX^k$ ba a polynomial of degree $n \gt 0$, and let $r\gt 0$. Suppose that $P$ is not the monomial $a_nX^n$, in other words there is at least an $i<n$ such that $a_i\neq 0$. ...
0
votes
1answer
40 views

Values of coefficients of polynomial

Suppose we have the polynomial on $\mathbb{C}$: $$p(z)=a_nz^n+a_{n-1}z^{n-1} + \dots + a_1z+a_0$$ and the factored form: $$p(z)=a_n(z-z_1)^{d_1}(z-z_2)^{d_2} \dots (z-z_r)^{d_r}, \sum_{i=1}^r{d_i} ...
2
votes
1answer
129 views

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.
4
votes
2answers
276 views

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 ...
2
votes
0answers
63 views

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 ...
3
votes
1answer
142 views

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 ...
1
vote
3answers
64 views

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 ...
1
vote
1answer
31 views

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 ...
4
votes
2answers
123 views

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 ...
2
votes
1answer
56 views

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 ...
6
votes
1answer
100 views

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 $P$ ...
2
votes
1answer
76 views

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 ...
0
votes
1answer
44 views

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 ...
1
vote
1answer
26 views

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 ...
1
vote
1answer
27 views

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 ...
1
vote
1answer
37 views

How rapidly can a polynomial grow in a proximity of the real segment comparing to the values on the segment?

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 ...
2
votes
1answer
53 views

Zeros of polynomials are continuous

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 ...