Tagged Questions

2
votes
1answer
51 views

Is there a geometric relationship between plane geometry and polynomials?

It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
0
votes
1answer
56 views

Holomorphic functions with polynomial real part

$f:\mathbb{C}\rightarrow \mathbb{C}$, $f(x+iy)=u(x,y)+iv(x,y)$ is a holomorphic function, its real part $u$ is a harmonic polynomial, i.e. $u\in \mathbb{R}[x,y]$ and $\frac{\partial^2 u}{\partial ...
2
votes
1answer
52 views

Rouché's Theorem on $z^{10} + 10z + 9$

Please note: this question was asked before, but the solution provided does not work as far as I know; see How to find the number of roots using Rouche theorem? We have $f(z) = z^{10} + 10z + 9$ and ...
1
vote
4answers
73 views

Calculating a complex derivative of a polynomial

What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials? For instance, is it justified to calculate the partial derivatives of ...
9
votes
1answer
239 views

Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
1
vote
0answers
23 views

Maximum modulus of complex polinomial in an open ball

Hi I'm stuck in this problem: Let $p\in \mathbb C[z], a\in \mathbb C$ and $r\geq 0$, then exists $z\in D_{r} (a)$ such that $|p(z)|>|p(a)|$. I'm trying to figure out what to do but I get nothing, ...
2
votes
1answer
72 views

how to find bounds on (complex) coefficients from bounds on a polynomial?

I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients: If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
0
votes
1answer
38 views

Zeros of the analytic limit of complex polynomials

For $n\in\mathbb{N}$ let $p_n$ be a polynomial of degree $n$. Suppose there exists $c>0$ such that $\bullet$ if $z\in\mathbb{C}$ is a zero of a $p_n$, then $|z^2+c|\leq c$ (note that in particular ...
5
votes
1answer
47 views

Roots of a polynomial and its derivative

All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part. It's not a homework. This issue has been proposed in the ...
5
votes
3answers
57 views

Determine complex polynomial

Problem Let $P(z) = z^n + a_{n−1}z^{n−1} + \cdots + a_1z + a_0$ be a polynomial of degree $n > 0$. Show that if $\lvert P(z) \lvert \le 1$ whenever $\lvert z \rvert = 1$ then $P(z) = z^n$. I ...
0
votes
1answer
62 views

Sum of all the residues of the function $a(z)/b(z)$

Let $a(z)$ and $b(z)$ be polynomials such that $ \deg(b) \ge \deg(a)+2$. Find the sum of all the residues of the function $a(z)/b(z)$. In class, I learned that $$ - \text{ sum of all residues of ...
4
votes
2answers
102 views

A ‘strong’ form of the Fundamental Theorem of Algebra

Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial $$ p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
0
votes
3answers
193 views

Proof real coefficients complex analysis

Show that if the polynomial $p(z)$ has real coefficients, it can be expressed as a product of linear and quadratic factors, each having real coefficients. I am not sure how to prove this. ...
5
votes
1answer
153 views

Complex Analysis - Location of roots of a polynomial

How many roots does the polynomial $z^4 + 3z^2 + z + 1$ have in the right-half complex plane (i.e. $Re(z) \gt 0$)? I honestly can't think of how to approach the problem as it seems different from the ...
1
vote
1answer
79 views

polynomial in several variable whose maximum modulus on the ball is known exactly

I'm interested in polynomials in several variables $p(x_1,\ldots,x_n)$, with complex coefficients, such that the maximum modulus of $p$ on the unit complex $n$-ball $$ \max \{ |p(z_1,\ldots,z_n)| : ...
0
votes
1answer
248 views

Coefficients of a cubic equation having one positive real root and two complex root with negative real part

Let $0 \lt \alpha \lt 1$ and $\beta,\gamma \gt 0$. Let $p(x) =x^{3}-\gamma x^{2}-\alpha x-\frac{\beta }{\gamma }$. Can we choose $\alpha ,\beta ,\gamma $ such that $p(x)$ has one positive real root ...
0
votes
0answers
40 views

Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.

We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$. Let us write $L(v) \equiv L(v,p)$ ...
1
vote
0answers
41 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
0
votes
0answers
36 views

Upperbound on the number of zeros of simple continuous non-polynomials

Let $[vf(p)-g(p)]$ be a degree $n$ polynomial, one of whose roots are $p = F(v)$. Note: We said $F(v)$ is one of the $n$ roots. Hence, it is a single valued and continuous function of $v$. Let us ...
2
votes
1answer
33 views

Sequence of polynomials (Q2)

Define $P_0(x) = 0$ and for $n > 0, \ P_n(x) = (x \ + \ P_{n-1}^2(x)) / 2$ and $Q_n(x) = P_n(x) - P_{n-1}(x)$. Are all the coefficients of the polynomials $Q_n(x)$ nonnegative?
1
vote
0answers
108 views

Factoring large polynomials

I'm asked questions like to find the zeros, and multplicities of the $\mathbb{C}$-polynomials $$f(z) =z^6 +4z^2 - 1 \hspace{10mm} , |z| < 1$$ or worse yet $$f(z)=z^{87}+36z^{57}+71z^4+z^3-z+1 ...
1
vote
1answer
138 views

Complex analysis: What is the mysterious polynomial?

I am given 2 pieces of information as below: 1)A polynomial $\displaystyle P(z)=\sum_{n=0}^d a_n z^n$ 2) For all $n=0,\dots,d$, $\displaystyle \oint_{|z|=1} \frac{P(z)}{(2z-1)^{n+1}} dz = 0 $ Then ...
1
vote
1answer
61 views

How zeros of second derivative map to equilibrium value

This question arose from plotting some functions in MATLAB and GeoGebra. Assume we have a function of the form $$f(x)=A \sin(P_n(x))+d \qquad \text{or}\qquad f(x)=A \cos(P_n(x))+d$$ where $$P_n(x) ...
2
votes
1answer
126 views

When does a higher order polynomial have complex roots?

I try to say it all in the title. I'm wondering under what conditions a matrix will have complex eigenvectors and eigenvalues. That question, I think, reduces to whether the characteristic ...
2
votes
1answer
37 views

$\int_{T^n}|P|=0$ implies $P=0$?

This is Exercise 1 of Chapter 8 in Rudin's Functional Analysis. We are asked to prove the following: If $P$ is a polynomial in $\mathbb{C}^n$ and if ...
1
vote
2answers
143 views

Complex analysis integration question

Let $f(z) = A_0 + A_1z + A_2z^2 + \ldots + A_nz^n$ be a complex polynomial of degree $n > 0$. Show that $\frac{1}{2\pi i} \int\limits_{|z|=R} \! z^{n-1} |f(z)|^2 dz = A_0 \bar{A_n}R^{2n}$.
5
votes
1answer
431 views

prove that the entire function f is a polynomial.

Suppose that $f$ is an entire function, and that in every power series $f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^n$ at least one coefficient is 0. Prove that $f$ is a polnomial. Hint: ...
-1
votes
1answer
137 views

How to find asymptotic entire functions?

I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...
3
votes
2answers
142 views

Trouble with representing power series as polynomials.

I am a math student trying to wrap my head around complex analysis through self-study. I am using Complex Analysis by Serge Lang, but I find myself struggling with some of his power series ...
6
votes
1answer
320 views

Zeros of a complex polynomial

The question is: Show that $$ P(z) = z^4 + 2z^3 + 3z^2 + z +2$$ has exactly one root in each quadrant of the complex plane. My initial thought was to use Rouche's Theorem (since that's generally ...
4
votes
1answer
88 views

Polynomial of same degree

Let $p(z)$ and $q(z)$ are two polynomial of same degree and zeroes of $p(z)$ and $q(z)$ are inside open unit disc, $|p(z)|=|q(z)|$ on unit circle then show that $p(z)=\lambda q(z)$ where ...
7
votes
1answer
194 views

When does an “infinite polynomial” make sense?

Suppose I pick a collection $A \subset \mathbb{C}$ of points in the complex plane and attempt to construct a "polynomial" with those roots via, $$f(z):=\Pi_{\alpha \in A} (z-\alpha).$$ If $A$ is ...
2
votes
1answer
73 views

Convergence to zero of a sequence

Let $c > 0$. I'm trying to show that the sequence $\displaystyle\sum\limits_{k=0}^n \left|\frac{n^k-\frac{n!}{(n-k)!}}{n^k}\right|\frac{c^k}{k!}$ converges to zero, as $n \to \infty$. I know that ...
3
votes
2answers
191 views

A question on the Fundamental Theorem of Algebra

I just read about a wikipedia page on Fundamental Theorem of Algebra, and it says "Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex ...
3
votes
3answers
285 views

complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$ inside balls

this is a question from a Temple prelim exam, and i'm trapped in it! We have $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$ and we have to prove that $\forall r>0 \quad \exists N\in\mathbb{N}$ s.t. $p_n(z)$ ...
0
votes
2answers
255 views

Polynomial expansion of complex numbers

Given $A$ and $B$ are complex numbers. I want to request anyone who might know any formulas for expanding this following expression. $$ |A-B|^{2n}$$ where $n$ is an integer. The one that I commonly ...
3
votes
1answer
285 views

Formula for Legendre polynomials by use of Cauchy's Integral Formula (From _Visual Complex Analysis_)

I decided to look through Tristan Needham's Complex Analysis book since it's usually mentioned with great praise. Just doing some exercises, I got stuck on #4 of Chapter 9). Here $P_n(z)$ denotes the ...
3
votes
1answer
135 views

Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
1
vote
2answers
191 views

Show that for any complex number W, there exists a non-zero complex number Z such that Z + 1/Z = W

Can anyone help me with this proof? I am not sure how to exactly go about this using just variables such as a + bi ? Thanks in advance!
1
vote
2answers
167 views

Complex Polynomial transformation

I'm studying for an exam and professor gave us to create a little program that automatically does a transformation for a polynomial with complex coefficients, I don't have many problems doing the ...
1
vote
1answer
42 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
4
votes
1answer
95 views

The zero set of sums of polynomials

As I am new to this forum, please correct me if this post is not appropriate. In that case I apologize. Let $P(z)$ and $Q(z)$ be polynomials with coefficients in $\mathbb{C}$, furthermore let $Z(P)$ ...
6
votes
1answer
190 views

Calculating $\prod (\omega^j - \omega^k)$ where $\omega^n=1$.

Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate $$ \prod_{j \ne k} (\omega^j - ...
2
votes
1answer
220 views

Bound the complex roots of a polynomial above

We consider $P(z)=a_{0}+a_{1}z+\cdot+a_{n-1}z^{n-1}+a_{n}z^n$, with $a_{0},\ldots,a_{n-1},a_{n} \in \mathbb{C}$ and $a_{n}\neq0$. Let $R=\max_{0\leq k\leq n-1}\left | \frac{a_k}{a_n} \right |$ and ...
4
votes
1answer
182 views

Find number of roots in some area (Rouché's theorem)

The task is to find number of $ {z^4} + {z^3} - 4z + 1 = 0$ in the area $1 < \left| z \right| < 2$. (this task is in the Rouché's theorem paragraph) I used this theorem many times, but I ...
4
votes
1answer
104 views

Polynomials that are orthogonal over curves in the complex plane

Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other ...
5
votes
1answer
111 views

Preimage of discs under a complex polynomial

Let $a_0, \ldots, a_n \in \mathbb{C}$, with $a_n \neq 0$. Consider set $$U_R = \{~z \in \mathbb{C} ~:~ |a_nz^n + \dots + a_1z + a_0| < R~\}$$ for each $R > 0$. How do I prove that $U_R$ is ...
2
votes
1answer
727 views

How many roots of a polynomial have positive real part?

I am given an exercise with three polynomials, and we have to find the number of roots of the first one that lie in the unit disk, the number of roots that lie in some region, e.g. those that lie in ...
24
votes
2answers
317 views

Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?

Consider the following spectacular image, created by Sam Derbyshire and described in John Baez's article "The Beauty of Roots": In this image are plotted all the complex roots of all polynomials of ...
0
votes
1answer
177 views

Polynomial mappings of half planes and disks on the complex plane

Main problem: Let $\mathcal{L}=\left\{z\in\mathbb{C}:Re(z)<0\right\}$ be the left open half-plane of the complex plane and $\mathcal{C}=\left\{z\in\mathbb{C}:|z|<1\right\}$ be the open unit disk ...

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