**1**

vote

**1**answer

47 views

### Showing that a particular area is small

Note: I posted this on math.stackexchange.com earlier (original post here: http://math.stackexchange.com/questions/1471331/showing-that-a-particular-area-is-small), but it received no responses and ...

**14**

votes

**1**answer

920 views

### Are the following identities well known?

$$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
...

**13**

votes

**2**answers

192 views

### Non-negative polynomials on $[0,1]$ with small integral

Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a_n = \min_{p\in P_n} \int_0^1 ...

**-3**

votes

**0**answers

30 views

### A question on subordinate matrix norm

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 ...

**4**

votes

**0**answers

49 views

### Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...

**0**

votes

**0**answers

43 views

### Construct a locally concave polynomial [on hold]

I'd like to construct a polynomial $f(x)$ such that:
$f(x)$ has roots 0 and $1$ (and possibly others)
$f'(a) = 0$ for given $a$, where $0 < a < 1$
$f(a) = 1$ (ie. there's a local maxima at ...

**2**

votes

**1**answer

72 views

### Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary ...

**1**

vote

**1**answer

70 views

### Existence of real solutions for a system of linear and quadratic equations

Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not ...

**21**

votes

**2**answers

432 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**7**

votes

**0**answers

94 views

### Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...

**8**

votes

**1**answer

110 views

### a generalization of gamma matrices

Is it possible to find matrix solutions to the following :
$$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$
where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) ...

**1**

vote

**0**answers

32 views

### Minimum degree of a nonnegative polynomial uniformly approximating two constant values on two disjoint closed intervals

This is a one-dimensional problem over $\mathbb{R}$. Given $y_0, y_1 \ge 0$ with $y_0 \neq y_1$, and closed intervals $I_0$ and $I_1$ with $I_0 \cap I_1 = \emptyset$, define a partial function $f(x) = ...

**8**

votes

**1**answer

237 views

### A characteristic 2 polynomial recursion

Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ be defined by the recursion $$c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions
$$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad ...

**8**

votes

**0**answers

98 views

### Order of zeros for sparse polynomials mod $p$

It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c
\neq 0$, $f$ has a zero of order at ...

**5**

votes

**0**answers

208 views

### Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...

**-1**

votes

**0**answers

49 views

### A question on boundary set

Suppose:
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, ...

**12**

votes

**0**answers

263 views

### Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...

**7**

votes

**1**answer

166 views

### Do semialgebraic sets depend outer semicontinuously on their defining polynomials?

Consider a compact (semialgebraic) ball $B\subset \mathbb{R}^n$ and a semialgebraic set $A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n$ defined through some representation in terms of polynomials ...

**-3**

votes

**0**answers

78 views

### Representations of integer in the form f(x) - f(y)? [migrated]

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. I would like to have an estimate for the number of representations $R(n)$ of $n \in \mathbb{Z}$ in the form
$$
f(x) - f(y) = n, \qquad x,y \in \mathbb{N}.
...

**5**

votes

**0**answers

108 views

### On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of ...

**5**

votes

**2**answers

526 views

### Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?

I asked this question at MSE but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by ...

**4**

votes

**1**answer

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

**6**

votes

**2**answers

386 views

### Why are most coefficients of these minimal polynomials divisible by $p$?

For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...

**5**

votes

**1**answer

156 views

### Determinant of symmetric Latin square

Let $n=2m$ be an even number. Let us construct $n\times n$ symmetric matrices $S_n$ in the following way. The entries are indeterminates $X_1,\ldots,X_{n-1}$. We choose a $1$-factorization of the ...

**5**

votes

**1**answer

402 views

### An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an ...

**2**

votes

**0**answers

61 views

### A question about polynomial over finite field

For the positive integer $m$, let $n=\varphi(m)$ denote the totient of $m$.
Given a rational prime $p$ such that $p\equiv 1 \bmod m$, let $A=\{a_1,\cdots, a_n\}\subseteq \mathbb{F}_p$ denote the set ...

**2**

votes

**0**answers

97 views

### Unique solution for a specific system of polynomial equations

Let $x^{(1)}, \dots, x^{(N)} \in \mathbb R^n$ be given. Suppose all we know about the $x^{(i)}$ are the values $c_1, \dots, c_N$ as given below
\begin{align*}
&c_1 := \sum_{i=1}^N \langle v^{(j)}, ...

**7**

votes

**0**answers

105 views

### Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...

**7**

votes

**2**answers

309 views

### Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination

I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}_{10} \left({x}\right) = \prod\limits_{1 \le i ...

**3**

votes

**1**answer

124 views

### How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques.
For example, consider the equation $qn = mf$, where each of the variables ...

**2**

votes

**0**answers

72 views

### Polynomial constraints triggered by irreducibility [closed]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic:
$$af^2 + bf + c = 0$$
If we're working in a ring, ...

**38**

votes

**5**answers

2k views

### Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g.,
$$z^3+z^2+2 z+3 \;.$$
Find its $n$ roots, and list them in order of their modulus:
$$-1.28, (0.14\pm 1.53 i)$$
Now form a new ...

**6**

votes

**1**answer

297 views

### Negative coefficient in an almost cyclotomic polynomial

Let $a,b,c,d$ be four prime numbers. We set the polynomial :
...

**2**

votes

**0**answers

64 views

### Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices

I'll give two equivalent statements of the setup, then give my questions.
Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in ...

**1**

vote

**0**answers

57 views

### Perturbed Chebyshev polynomials

It is well-known that the Chebyshev polynomials of the first kind satisfy the recurrence relation
$$
\begin{cases}
T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\
T_{0}(x)=1, \ \ T_{1}(x)=x \\
...

**11**

votes

**1**answer

101 views

### Commuting ODE's implies existence of nonzero vanishing two variable polynomial?

Write $\partial := d/dt$, fix $m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ...

**0**

votes

**0**answers

29 views

### Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...

**1**

vote

**0**answers

29 views

### About the partial expectation polynomials in the Interlacing-I paper and perfect matchings

I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf
In the use of these ...

**6**

votes

**0**answers

93 views

### Recursions which define polynomials?

Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...

**16**

votes

**2**answers

803 views

### Which algebraic relations are possible between algebraic conjugates?

For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More ...

**4**

votes

**0**answers

168 views

### An equivalent 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 ...

**5**

votes

**2**answers

204 views

### Dickson/determinant type polynomial (updated)

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}
...

**6**

votes

**1**answer

176 views

### Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in ...

**9**

votes

**2**answers

204 views

### A generalization of Chebyshev polynomials

What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$.
Now suppose ...

**2**

votes

**1**answer

185 views

### Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
2\cos\frac{2j\pi}{n} & -m \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & ...

**1**

vote

**0**answers

133 views

### An identity satisfied by “Differentiation”

I asked this question in MSE but I did not received any answer. So I repeat it here:
Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...

**36**

votes

**3**answers

1k views

### Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that
$$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$
for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...

**1**

vote

**0**answers

61 views

### degree of associative algebra

Let $A$ be a finite dimensional associative algebra with unity over a field $F$. The degree of the algebra is the degree of its generic minimum polynomial (see Nathan Jacobson, Generic norm of an ...

**6**

votes

**0**answers

171 views

### Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...

**5**

votes

**2**answers

325 views

### Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.
What is the isomorphism class of the inertia group $I_p$, ...