**1**

vote

**0**answers

20 views

### A good version of truncated real radical ideal?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...

**6**

votes

**1**answer

209 views

### On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for ...

**0**

votes

**0**answers

152 views

### Warren's Theorem

At the end of page 12 in this document Noga Alon mentions Warren's Theorem on sign patterns: tau.ac.il/~nogaa/PDFS/tools1.pdf
Does anyone know of an intuitive explanation of the proof of it ? Also, ...

**1**

vote

**1**answer

107 views

### On a reciprocal of Ostrowski theorem on Newton polytopes and factorization

$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...

**0**

votes

**0**answers

29 views

### Partial Fraction Decomposition based on Monomials

Given two positive integers $m$ and $k$, complex $a$ and the rational polynomial
$$ q(z) = \frac{1}{z^{m+k} + a z^{m} + a z^{k} + 1}. $$
Is there a partial fraction expansion over the complex numbers, ...

**-5**

votes

**0**answers

71 views

### Can any polynomial expressed as a chromatic polynomial? [on hold]

Is it possible any polynomial with integer coefficients to be (or converted to) a chromatic polynomial that corresponds to a graph?

**3**

votes

**1**answer

117 views

### A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let ...

**5**

votes

**1**answer

355 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 $A \in \mathbb R^{m \times 5}$ ...

**4**

votes

**1**answer

129 views

### Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...

**10**

votes

**3**answers

1k views

### When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.)

As a natural (and expectable) extension of my earlier question:
How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a cubic polynomial in $n$ variables over the field $F_2$, ...

**2**

votes

**0**answers

133 views

### On reducible polynomials

Let $f(x),g(x)\in\Bbb Z[x]$ with $deg(f)>deg(g)$.
Given an integer $B$, is there any algorithm that runs in $\log^c |B|$ for some fixed $c\in \Bbb R$ to find a $h(x)\in\Bbb Z[x]$ (if one exists) ...

**9**

votes

**0**answers

86 views

### Factorisation in $\mathbb{N}[X]$?

Do we know an efficient algorithm to factorise in $\mathbb{N}[X]$ ?
One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and to combine some factor to ...

**5**

votes

**1**answer

78 views

### Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set ...

**1**

vote

**3**answers

744 views

### Detecting if a polynomial is a Pfaffian

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?
The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ ...

**2**

votes

**0**answers

47 views

### Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form
$$K(n,m)=\frac{2^{m+n} q^{(m+n+1)}}{n-m} \left[\Gamma(\frac{n}{2}+1)\sin(\pi \frac{n}{2})\Gamma(\frac{m-1}{2}+1) \sin(\pi \frac{m-1}{2}) - ...

**20**

votes

**5**answers

2k views

### Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?

The question
Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...

**4**

votes

**2**answers

186 views

### Invariant polynomials under diagonal action of the orthogonal group

Consider the diagonal action of the orthogonal group $O(n)$ on $\mathbb{R}^n\times\mathbb{R}^n$ defined as: $U\cdot (x,y) = (Ux,Uy)$ for $U\in O(n)$ and $x,y\in\mathbb{R}^n$. I am looking for a ...

**4**

votes

**1**answer

204 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

**20**

votes

**1**answer

882 views

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

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

**0**

votes

**0**answers

29 views

### Does this system have a closed-form solution? $x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $

I am interested in solving the following system of $n$ equations:
$$x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$
for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, ...

**13**

votes

**2**answers

455 views

### regular polygon question

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

**1**answer

76 views

### Octahedron and System of trigonometric equations

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

**2**

votes

**0**answers

65 views

### Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...

**4**

votes

**0**answers

47 views

### Is the restriction of a graded automorphism linearizable in characteristic zero?

This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...

**13**

votes

**3**answers

1k views

### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" ...

**7**

votes

**2**answers

331 views

### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...

**4**

votes

**1**answer

175 views

### Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...

**2**

votes

**1**answer

67 views

### Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...

**3**

votes

**1**answer

89 views

### Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points.
Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...

**3**

votes

**0**answers

100 views

### When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k ...

**1**

vote

**0**answers

59 views

### Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$
Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...

**5**

votes

**0**answers

69 views

### Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group

Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...

**1**

vote

**1**answer

92 views

### Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis.
Suppose there is a matrix
$$
A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] ...

**5**

votes

**2**answers

171 views

### Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...

**8**

votes

**1**answer

570 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 ...

**5**

votes

**4**answers

394 views

### How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form:
$$
(x-a)(x-b)(x-c)=d(x-e)(x-f),
$$
where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable.
Of ...

**3**

votes

**0**answers

115 views

### Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity:
$$ P(x)= \left( \sum_{\alpha \in P} ...

**12**

votes

**2**answers

969 views

### About irreducible trinomials

This question was inspired by this one. For every $n>m>0$ consider the polynomial $p_{m,n}=x^n-x^m-1$.
For which $m,n$ is $p_{m,n}$ irreducible over $\mathbb Q$?
In particular, if $m$ is ...

**3**

votes

**0**answers

162 views

### Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix
$$A = \left(\begin{array}{}
1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots ...

**3**

votes

**1**answer

118 views

### How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
...

**2**

votes

**0**answers

112 views

### Values of Bernoulli polynomials at roots of unity

I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.

**3**

votes

**1**answer

188 views

### system of complex equations

I am working on a system of complex equations The question is the following:
Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that
$$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} ...

**4**

votes

**0**answers

103 views

### Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$
I am ...

**0**

votes

**0**answers

33 views

### Comparing product of positive affine functions over integers

Problem
Let $f_i: \mathbb Z^n \mapsto \mathbb Z$ and $g_i: \mathbb Z^n \mapsto \mathbb Z$ affine functions and $\mathcal D \subseteq \mathbb Z^n$ a set on which they are all positive. Let $P$ and $Q$ ...

**5**

votes

**2**answers

161 views

### $L^{\infty}$ polynomial approximation

In short: For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial?
Jackson (1911) proved that there is a best approximating polynomial in the ...

**3**

votes

**1**answer

129 views

### Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive
polynomial $$F(x) = ...

**8**

votes

**0**answers

537 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
\begin{align}
P_{0,0}&=1\\
\text{for $n\geq ...

**3**

votes

**0**answers

98 views

### An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...

**3**

votes

**1**answer

200 views

### Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let ...

**2**

votes

**0**answers

60 views

### Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write ...