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)| : ...
