All Questions
Tagged with cv.complex-variables polynomials
152 questions
20
votes
1
answer
616
views
Conjecture on the number of roots of $z^n + P(z)$ within the unit disk
Some other people and I have noticed that the following seems to be true.
Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit ...
- 303
7
votes
2
answers
186
views
Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 3,317
2
votes
1
answer
115
views
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
- 187
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
0
votes
1
answer
116
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
- 4,058
0
votes
1
answer
103
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
- 826
2
votes
1
answer
133
views
Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients
I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
- 826
2
votes
1
answer
194
views
Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
- 3,319
1
vote
2
answers
151
views
Location of the negative real roots of certain integer-valued polynomials
The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a ...
- 1,889
2
votes
0
answers
164
views
Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
- 4,058
3
votes
1
answer
190
views
Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?
It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.
Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
- 4,058
0
votes
0
answers
160
views
Gegenbauer polynomial relation with complex argument
Gegenbauer polynomials, $C_j^{\nu}(t)$, are defined to be the coefficient of $h^j$ in the expansion $(1-2ht +h^2)^{-\nu}$. It can be shown using [Higher Transcendental Functions, Vol 1, Harry Bateman, ...
- 57
5
votes
0
answers
159
views
Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
- 5,230
11
votes
3
answers
726
views
Can computers find zeros of order $2$?
We assume we are given an entire function $f: \mathbb C \to \mathbb C$ with $f(0)=1$ and $f'(0)=0$ and $f$ is real on the real axis.
We assume (as a fact about $f$, that we want to demonstrate ...
- 124
3
votes
1
answer
135
views
On well separated circular regions in the Riemann sphere and complex polynomials
It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...
- 5,215
5
votes
1
answer
833
views
A statement on complex polynomials
I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...
- 5,215
20
votes
2
answers
729
views
Multiple roots of polynomials with coefficients $\pm 1$
Question P. Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?
Also I am interested in a similar question ...
- 41.8k
1
vote
1
answer
204
views
Complex polynomial-like functions with conjugate terms
Is there study on polynomial-like functions of the following kind?
$$f(z) = c_0 + a_1z+b_1\bar{z} + a_2z^2+b_2\bar{z}^2 + ...+ a_nz^n+b_n\bar{z}^n$$
My reason for studying it is polynomials are ...
- 175
7
votes
2
answers
907
views
Product of complex numbers on the unit circle with largest real part
Let $T = \{z_1, \ldots z_n\}$ be a finite set of complex numbers on the unit circle. I would like an algorithm which can quickly compute the nonempty subset $S \subset T$ which maximizes $$\left| \...
- 1,703
1
vote
0
answers
63
views
Bound on the coefficient of a $P$-adic decomposition of a polynomial in term of the sup norm
Let :
$P=\prod_{i=1}^d(T-\alpha_i)\in \mathbb C[T]$ such that for $i\neq j$, $\alpha_i\neq\alpha_j$ ;
$Q=\sum_{k=0}^nQ_kP^k\in\mathbb C[T]$ with $\forall k,\ \deg(Q_k)\leqslant \deg(P)-1$ be the $P$-...
4
votes
2
answers
376
views
$n-1$ quadratic forms for $n$ variables
If we have $n-1$ quadratic forms for $n$ variables $x_i$,
$$p_i(x) = M^{(i)}_{jk} x_j x_k$$
for $1\leq i \leq n-1$ and $1 \leq j,k \leq n$ then the zeros of all $p_i(x)$,
$$p_i(x) = 0$$
is generically ...
- 1,150
3
votes
1
answer
137
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
- 463
1
vote
2
answers
588
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
4
votes
0
answers
179
views
As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?
As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
8
votes
0
answers
199
views
Roots of a family of polynomials forming shapes
Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$.
Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$.
The roots of $F_n$ seems to form "shapes&...
- 9,501
10
votes
1
answer
703
views
Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots
Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$.
What is known about ...
- 121
3
votes
0
answers
226
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
- 429
5
votes
1
answer
169
views
Efficients method for finding a zero of a multilinear complex polynomial in an specified region
Let P be a given multilinear polynomial in $\mathbb{C}[z_1,\dots,z_n]$ and $D\subset \mathbb{C}$ be a given disc in the complex plane. Does there exist an efficient method for checking that $P$ has a ...
- 4,484
7
votes
0
answers
203
views
Polynomials having all zeros in the closed left half plane
Let $$P(z) =\sum_{k=0}^n(\alpha_k+e^{i\gamma}\beta_k)z^k=P_1(z)+e^{i\gamma}P_2(z)$$ be a polynomial of degree $n$ with $\alpha_k, \beta_k\geq 0$ for $0\leq k\leq n, $ where $$P_1(z) =\sum_{k=0}^n\...
- 559
5
votes
0
answers
225
views
Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
- 7,367
7
votes
1
answer
378
views
Local optimum for Sendov's conjecture
For Sendov's conjecture, the distance 1 appears in the conjecture is tight, if one consider the polynomials $f_{n}(z) = z^{n} - 1$ for all $n\geq 2$. I wonder if this polynomial is the local optima ...
- 2,215
20
votes
1
answer
744
views
Refinement of mean value conjecture for complex polynomials?
I was playing around with Smale's Mean Value Conjecture and found a curious formulation of it which would be stronger (and which may simply be false). It seems to hold for `generic' random polynomials ...
7
votes
0
answers
169
views
Limiting behavior of a sequence of polynomials
Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line
$\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary
fact (equivalent to Lemma 9.13 here) that
if $u\in\mathbb{C}$ and $|u|=1$, ...
- 50.8k
1
vote
1
answer
341
views
A mysterious expression from a discriminant
I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of ...
- 41
1
vote
0
answers
124
views
Context for this discrete Cauchy integral formula
Notation: I will use the following conventions for discrete Fourier transforms (DFT) and discrete time Fourier transforms (DTFT):
$$\mathcal{D}_N[x_j](k) := \sum_{j=0}^{N-1} e^{-2\pi i j k} x_j$$
$$\...
- 956
6
votes
0
answers
753
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
- 41
3
votes
0
answers
233
views
Sequence unifomly bounded
Let $f(\lambda,z)$ be a continuous function on $\Bbb R^2$ such that
I) For $n\in\Bbb N$ and $x\in\Bbb R^*_+$ we have : $f(n,x)=\cos(nx)+ x O\big(\frac{1}{n}\big)$ as $n\to\infty$ and $x\in[n^{-1}\...
- 81
-1
votes
1
answer
87
views
Inferring polynomial rate of convergence from polynomial bound
Let $x_n$ be a non-negative valued sequence and suppose that the following hold:
$\lim\limits_{n\to\infty} x_n =0$
There exists some polynomial function $p$ of degree at-least $1$ such that:
$$
\|x_n\...
- 5,407
1
vote
0
answers
201
views
Is there a natural topology on $\mathbb{C}(t)[x_1,\ldots, x_n]$ with this property?
Is there a good topology on $A=\mathbb{C}(t)[x_1,\ldots, x_n]$ so that $A$ is a topological algebra with the following property:
For any $N>0$ and a polynomial $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
- 291
0
votes
0
answers
109
views
The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
2
votes
1
answer
120
views
Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$.
Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of
$$p(w)=n+\sum_{j=1}^{m}\frac{...
- 5,474
3
votes
0
answers
187
views
No common roots of complex polynomial and of its derivative
Our specific context
Here is our specific contour integral
$$\int_{\Gamma_{0}}F\big(\sum_{w:p_{z}(w)=0}\frac{1}{w^{a}}\frac{1}{n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}} \big)\frac{dz}{z},$$
...
- 5,474
4
votes
1
answer
544
views
A problem on polynomials
Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$
- 559
4
votes
4
answers
514
views
Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?
let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(...
4
votes
1
answer
395
views
A possible generalization of Gauss Lucas theorem to higher dimension
A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $...
- 356
7
votes
1
answer
244
views
Volume of solution sets for polynomials in $\mathbb{C}[x]$
Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set
$$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\...
- 43.1k
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
3
votes
0
answers
116
views
Another application of Borel-Cantelli Lemma
I ask this question on math stackexchange, but there is no answer, so please forgive me I ask it here again.
Let $c>0$ and $P(x)$ be a polynomial. Then there exists a $p>1$ (e.g. we can take $p$...
- 61
1
vote
0
answers
62
views
How to prove the rational polynomial is nonincreasing?
Suppose $p(x)$ is a univariate real-rooted polynomial. It is easy to see that the following rational polynomial
$$\Psi_p(x) = \frac{\partial^2p}{p}(x)=\sum_{1\leq j<k\leq r}\frac{2}{(z_i-\...
- 131
2
votes
0
answers
180
views
Multiple zeta values related to fractional calculus and an Appell polynomial sequence
There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
- 10.5k