**12**

votes

**0**answers

297 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

200 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 ...

**5**

votes

**0**answers

123 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

546 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 ...

**6**

votes

**1**answer

338 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

402 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 $\xi$ ...

**6**

votes

**1**answer

170 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 ...

**6**

votes

**1**answer

484 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

77 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

120 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

121 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

328 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

175 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

76 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, ...

**39**

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

308 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

69 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

69 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 \\
...

**15**

votes

**1**answer

133 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

32 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 ...

**2**

votes

**0**answers

34 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

96 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

828 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 ...

**6**

votes

**1**answer

502 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 ...

**5**

votes

**2**answers

223 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

184 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 ...

**10**

votes

**2**answers

243 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

194 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

145 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 ...

**54**

votes

**6**answers

2k 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

65 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

184 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

370 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 $I_p$, ...

**5**

votes

**1**answer

418 views

### Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let ...

**5**

votes

**0**answers

162 views

### Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...

**6**

votes

**1**answer

413 views

### Is every polynomial a factor of a trinomial?

We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each ...

**0**

votes

**0**answers

112 views

### Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients
takes infinitely many prime values unless there are obvious reasons not
to.
It appears to imply something about polynomial ...

**1**

vote

**0**answers

29 views

### Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...

**9**

votes

**0**answers

192 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 ...

**13**

votes

**2**answers

588 views

### Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...

**3**

votes

**1**answer

227 views

### Polynomial with the smallest area

Let $P_n(t) = p_0 + p_1 t + \cdots + p_n t^n$ be a polynomial (with real coefficients) of degree $n$ in the variable $t$. I am interested in the quantity $$\Phi_n = \min_{\sum_{i=1}^n p_i^2 = 1} ...

**9**

votes

**3**answers

491 views

### Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...

**6**

votes

**1**answer

117 views

### Polynomial interpolation of binary word signal

Let consider a binary word $x_1 \ldots x_n$ (finite sequence of elements of $\{0,1\}$.
I want to construct a polynomial $P$ that interpolates the points $(i, x_i)$ for $i \in \{1\ldots n\}$ , such ...

**1**

vote

**1**answer

248 views

### Solutions to system of polynomial equations over finite fields

If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower ...

**11**

votes

**1**answer

265 views

### When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor?

When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor? Is there a general approach for any number of variables, aka when is the variety defined by the ...

**8**

votes

**3**answers

489 views

### Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...

**5**

votes

**4**answers

402 views

### Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:
Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...

**3**

votes

**0**answers

233 views

### When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...

**3**

votes

**1**answer

137 views

### Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...

**7**

votes

**3**answers

325 views

### Generalized Characteristic Polynomial with Unimodular Roots

Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{m_1}, \dots, z^{m_N})$ with $z\in\mathbb{C}$ and positive integers $m_1, \dots, m_N$.
The generalized characteristic polynomial of a matrix ...