**5**

votes

**1**answer

120 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 ...

**25**

votes

**5**answers

1k 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 ...

**2**

votes

**0**answers

62 views

### Polynomial constraints triggered by irreducibility [on hold]

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, ...

**3**

votes

**0**answers

58 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 ...

**4**

votes

**1**answer

155 views

### Negative coefficient in an almost cyclotomic polynomial

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

**9**

votes

**2**answers

251 views

### Can Schwartz-Zippel be formulated for commutative rings instead of fields?

The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for ...

**10**

votes

**1**answer

90 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 + ...

**2**

votes

**0**answers

49 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 ...

**13**

votes

**4**answers

628 views

### Finite interpolation by a nondecreasing polynomial

Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences
of $n$ real numbers. It is well known that there are polynomials that "interpolate"
in that ...

**1**

vote

**0**answers

50 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 \\
...

**6**

votes

**1**answer

57 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**

votes

**0**answers

56 views

### Interpolating Product of two Polynomials

Consider we have two non-constant polynomials $A(x)$ and $B(x)$. We define the polynomials over field $\mathbb{Z}_p$, for a large prime number $p$.
We define Polynomial $A(x)$ as follows: ...

**1**

vote

**0**answers

63 views

### Interpolating a Polynomial Given Multiplier of each $y_i$

It would be great even if you answer only one of the below questions.
We have polynomial $P(x)=(x-\beta)\cdot g(x)$, where degree of $P(x)$ is fixed n-1, $\beta$ chosen uniformly at random from the ...

**0**

votes

**0**answers

27 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

44 views

### Two rational and one irrational root of a cubic? [closed]

Let $p(x)=a_3x^3+a_2x^2+a_1x+a_0$, with $a_i\in\mathbb{Q}$. Is it true that if two of the roots of $p(x)$ are in $\mathbb{Q}$, then the third is as well?

**16**

votes

**2**answers

762 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 ...

**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 ...

**2**

votes

**1**answer

163 views

### About distinct eigenvalues of a graph

if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that ...

**6**

votes

**0**answers

86 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 ...

**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 ...

**2**

votes

**1**answer

181 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 & ...

**4**

votes

**0**answers

158 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

192 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}
...

**9**

votes

**2**answers

199 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 ...

**43**

votes

**6**answers

3k 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 ...

**16**

votes

**0**answers

394 views

### Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...

**1**

vote

**0**answers

131 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 ...

**1**

vote

**0**answers

60 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 ...

**4**

votes

**1**answer

617 views

### Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...

**36**

votes

**4**answers

2k views

### The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...

**12**

votes

**2**answers

498 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 ...

**6**

votes

**0**answers

166 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

321 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 $I_p$, ...

**4**

votes

**1**answer

364 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 ...

**4**

votes

**0**answers

140 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 ...

**5**

votes

**1**answer

377 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 ...

**-2**

votes

**0**answers

89 views

### Grade 7: When is an Algebraic Expression NOT a Polynomial? [migrated]

Ok so this appeared in our school, and we all didnt know the answer. We arent sure, but we think the answer is, "when there are Signs like the Square Root signs, Negative Integers, etc." Or "When a ...

**0**

votes

**0**answers

66 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 ...

**0**

votes

**0**answers

16 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 twodimensional case the ...

**11**

votes

**2**answers

372 views

### Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...

**4**

votes

**0**answers

51 views

### 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 ...

**8**

votes

**1**answer

307 views

### When adding a constant makes a multivariate polynomial reducible?

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?
...

**3**

votes

**1**answer

207 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

462 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}
...

**7**

votes

**3**answers

236 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 ...

**2**

votes

**1**answer

104 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 ...

**7**

votes

**3**answers

383 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 $p$) ...

**11**

votes

**1**answer

222 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 ...

**1**

vote

**1**answer

164 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 ...

**7**

votes

**4**answers

577 views

### What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?

(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
...