# Tagged Questions

**1**

vote

**1**answer

192 views

### Polynomial convex coefficients

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...

**4**

votes

**0**answers

61 views

+50

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial
is a bivariate polynomial with positive integer coefficient which is a graph
invariant and can be defined recursively.
Evaluating it is $\#P$-complete even when restricted to ...

**1**

vote

**0**answers

58 views

### Boundary of pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**-2**

votes

**0**answers

22 views

### Find the number of connected components in pseudospectra [on hold]

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**5**

votes

**0**answers

99 views

### Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...

**12**

votes

**6**answers

2k views

### English reference for a result of Kronecker?

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:
Let $f$ be a monic polynomial with integer ...

**0**

votes

**0**answers

54 views

### “Dimension” of ideals in $F_q[x]/\langle x^n-1\rangle$? [closed]

I'm very much confused by algebra. Hoping to get a bit more comfortable I tried to compute different things and see what happens...
Let $F_q$ be the finite field with $q$ elements and ...

**4**

votes

**1**answer

176 views

### Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$,
we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$
and $T$ be sets of elements of $\mathbb{F}$ of size ...

**0**

votes

**1**answer

79 views

### approximation of products of polynomials

I am wondering whether the following can be proved:
Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...

**18**

votes

**1**answer

489 views

### Why can the general quintic be transformed to $w^5-5\beta w^3+10\beta^2w-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic,
$$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...

**-1**

votes

**0**answers

153 views

### Where I can find a Carlitz's paper? [closed]

I am looking for a PDF of the paper:
Carlitz, Lewis, Mills, Straus - Polynomials over finite fields with minimal value sets.
Only I can find about it is this data:
Mathematika / Volume 8 / Issue ...

**1**

vote

**1**answer

157 views

### Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...

**5**

votes

**2**answers

1k views

### How to solve a fifth degree polynomIal

Charles Hermite have created a method using elliptic functions to solve fifth degree polynomial, to get around the theory of Galois. Can someone explain me it and give a simple example?
Tank you.

**1**

vote

**1**answer

107 views

### When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...

**0**

votes

**1**answer

95 views

### fixed points of an affine polynomial automorphism

Let $K:- k[x_1, x_2, \cdots, x_n]$ be the polynomial ring over a field $k$. Let $a_i, b_i \in K$ where $a_i \ne 0$. Consider the automorphism $\alpha$ of $K$ defined by
$x_i \mapsto a_ix_i + b_i$. ...

**5**

votes

**2**answers

104 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...

**2**

votes

**0**answers

241 views

### Fixed points of self maps

Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...

**47**

votes

**6**answers

4k views

### How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let ...

**-2**

votes

**2**answers

3k views

### Roots of a polynomial in several variables [closed]

Hi,
I'm new here, and not a mathematician at all :-(
I am looking for an algorithm to find the roots (in the complex domain) of a polynom of several variables.
Thanks for any light you could bring ...

**3**

votes

**1**answer

184 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**9**

votes

**0**answers

179 views

### On shifted symmetric power sums

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define ...

**0**

votes

**0**answers

33 views

### finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...

**4**

votes

**1**answer

173 views

### Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields?
For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...

**1**

vote

**1**answer

106 views

### Is there a limit definition for the roots of a polynomial with arbitrary degree? [closed]

I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest ...

**30**

votes

**3**answers

937 views

### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...

**0**

votes

**0**answers

48 views

### Is there any way to approximate the largest root of a polynomial? [duplicate]

I know Newton's method will always give a root, but is there a modified version of the method that will always give the largest root?

**2**

votes

**0**answers

110 views

### Polynomials with some roots whose product is 1

I asked this question in this post but have not got a full answer. So I post it again on MO.
Consider the complex coefficient polynomial equation
\begin{eqnarray}
...

**6**

votes

**1**answer

483 views

### An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**1**

vote

**1**answer

89 views

### Ternary cyclotomic polynomials with $n=15r$

Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily ...

**5**

votes

**2**answers

134 views

### Generalized cycle index polynomial for the symmetric group

The answer to a particular calculation in quantum information theory gives me the following expression:
Given $M$ specific elements of the symmetric group $S_n$, define the polynomial
$$Z_n(\pi_1, ...

**2**

votes

**0**answers

92 views

### Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...

**2**

votes

**1**answer

200 views

### On the divisibility of a certain power sum

Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?.
For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect ...

**5**

votes

**1**answer

259 views

### Injectivity of a multivariate homogeneous polynomial mapping

Consider the mapping
$$ \Psi: \mathbb R^2 \to \mathbb R^5, \\
\Psi(x) = \begin{pmatrix} x_1 \\ x_2 \\ x_1^2 \\ x_1 x_2 \\ x_2^2 \end{pmatrix}.$$
Which are the matrices $A \in \mathbb R^{m \times 5}$ ...

**0**

votes

**0**answers

14 views

### Complexity of computing the multivariate Tutte polynomial of clique where each edge have distinct label

The multivariate Tutte polynomial $Z_G(q,v)$
is generalization of the Tutte polynomial and each edge is labelled by
variable $v_e$.
$Z_G(q,v)$ is linear in $v_i$.
Let $G$ be a clique where each edge ...

**1**

vote

**0**answers

19 views

### Complexity of computing the Tutte polynomial of multigraph when the Tutte polynomial of the underlying simple graph is known

Let $G$ be multigraph with $l$ loops and $m$ multiple edges and $G'$ be the
underlying simple graph (loops and multiple edges removed).
Assume the Tutte polynomial of $G'$ is given.
Q1 What is ...

**1**

vote

**1**answer

163 views

### Reduced resultant of monic polynomials

Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the ...

**4**

votes

**0**answers

150 views

### Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...

**2**

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**0**answers

138 views

### Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...

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**0**answers

360 views

### Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...

**7**

votes

**1**answer

309 views

### Is the evaluation of polynomial functors appropriately continuous?

I'd like a nice proof of the following fact.
Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to ...

**0**

votes

**1**answer

113 views

### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...

**0**

votes

**2**answers

341 views

### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...

**0**

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**0**answers

41 views

### Approximation by polynomials with coefficients sum

In this paper Erdos p.1176 remarked that if the coefficients of $f(z):\sum_{v=0}^{n}a_{v}z^v$ are all real,then $\sum_{v=0}^{n}|a_{v}|$ is maximal for $f(z)=\pm T_{n}(z)$,where $T_{n}(z)$ is Chebyshev ...

**2**

votes

**1**answer

67 views

### Distribution of values of quadratic polynomials over a finite field

Let $S$ denote the set of squares in the finite field ${\mathbb F}_p$. Now
let $f(x) \in {\mathbb F}_p[x]$ denote a degree $2$ polynomial, which is not a
square of a linear polynomial. If we consider ...

**19**

votes

**0**answers

307 views

### Zero curves of Tutte Polynomials?

There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...

**7**

votes

**2**answers

309 views

### regular polygon and constant potential function

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**2**

votes

**0**answers

86 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
...

**5**

votes

**1**answer

199 views

### Estimating size of greatest prime divisor of a sequence of integers

Consider the numbers of the form: $$A_n = \prod_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$
where, the product in taken oven all $2^n$ terms with variations in sign. We know such ...

**3**

votes

**1**answer

144 views

### Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited:
Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...

**1**

vote

**0**answers

54 views

### Octahedron and System of trigonometric equations

Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
...