# Tagged Questions

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### Generalization of the Hermite-Beihler-Kakeya Theorem (2)

This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments.
Firstly we remark that: $f(x)+g(x)\cdot w$ is ...

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**2**answers

132 views

### Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where
${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...

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**1**answer

513 views

### Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...

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**1**answer

259 views

### Minimize norm of a polynomial around a circle (count the solutions)

I already posted this question at MSE here, but as it received no significative feedback for a while I cross-post it here.
I also noticed a related question here on MO (which does not answer my ...

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206 views

### The Poisson-kernel in the plane and polynomials

Let
\begin{align*}
p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\
& = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j}
\end{align*}
be a non-constant complex polynomial with ...

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**2**answers

331 views

### The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...

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**4**answers

723 views

### Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...

**22**

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**1**answer

1k views

### Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
...

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**1**answer

497 views

### Polynomial with all zeros on a circle and many real coefficients

On a circle (or a line) $\Omega$ in the complex plane that is not symmetric w.r.t. the real axis, choose $n\ge5$ distinct points $z_1,...,z_n$ and consider the polynomial ...

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242 views

### Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of ...

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**1**answer

306 views

### Integer polynomials mapping the unit disk into itself

Is there a complete characterization of those integer polynomials, that is $P\in{\mathbb Z}[X]$, such that $P(D)\subset D$, where $D$ is the unit disk ? At least, $P(X)=\pm X^k$ works, when ...

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**0**answers

427 views

### Possible application of Rouche's theorem to aproblem of complex roots of polynomials

The following holds:
Let $P(x)$ be a polynomial in one variable $x$ of degree $3$ with complex coefficients
such that
a)
$$
P(-1)=P(1)=0
$$
Then
b)
the formal derivative $P^{'}(x)$ has a root in ...

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**4**answers

2k views

### The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...

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**2**answers

356 views

### Bivariate polynomials with special properties

I recently came across some polynomials with some remarkable properties.
A polynomial $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if
the set of solutions to the system ...

**0**

votes

**1**answer

232 views

### Average compared with discrete average for some $\lbrace -1,1 \rbrace$ polynomials

Let $k>0$ be a positive integer. Set $n=4k.$ Let $R(t)$ be a polynomial of degree $n-1$
with coefficients in $\lbrace -1,1 \rbrace$.
Consider the discrete average
$$
D(n,R) = ...

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votes

**1**answer

499 views

### on common fixed points of commuting polynomials (and rational functions)

By the Ritt's classification, for any pair of commuting polynomials (i.e. $f(g(z))=g(f(z))$) over $\mathbb C$ there is a common fixed point of them. My questions are:
Is that true that this can be ...

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**2**answers

452 views

### Can the unit complex 1-dimensional disc be embedded isometrically into complex euclidean space?

Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed ...

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1k views

### Minimizing the modulus of a polynomial around a circle

I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...

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**2**answers

1k views

### Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.
First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...

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**1**answer

861 views

### How to best distribute points on two concentric circles?

An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le ...

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2k views

### bounding roots of a polynomial with Rouche's Theorem

Suppose f(z) = z^n - k [ z^(n-1) + ... + z + 1 ] where n is a positive integer and k is a real constant such that nk<1.
I have shown that a root of this ...

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**1**answer

549 views

### Bernstein inequality for multivariate polynomial

Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$.
So, are there ...