Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ...

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12
votes
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
5answers
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
1answer
308 views

Negative coefficient in an almost cyclotomic polynomial

Let $a,b,c,d$ be four prime numbers. We set the polynomial : ...
2
votes
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
6answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
3answers
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
4answers
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
0answers
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
1answer
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
3answers
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 ...