# Tagged Questions

**2**

votes

**1**answer

126 views

### A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...

**7**

votes

**0**answers

167 views

### Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...

**2**

votes

**2**answers

361 views

### Looking for ways how to calculate $\Phi_n(i)$

$Φ_n(1)$ and $Φ_n(−1)$ for the cyclotomic polynomials are well-known.
I am now looking for
$$Φ_n(i)$$
and/or
$$Φ_n(−i)$$
with i the complex unit.
At this moments my endeavours result in intricate ...

**1**

vote

**0**answers

128 views

### q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity.
Show that the polynomial ...

**0**

votes

**0**answers

64 views

### Must the radical of polynomial evaluated at integers be small enough at least once?

Basically I am interested if the radical of polynomial evaluated at integers can be small enough at least once.
Let $f \in \mathbb{Z}[x], \deg(f)>1$ be squarefree. For integer $a$ and $f(a) \ne 0$ ...

**1**

vote

**0**answers

292 views

### When is a cubic polynomial a cube? [closed]

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers.
$$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...

**6**

votes

**4**answers

844 views

### Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...

**9**

votes

**2**answers

445 views

### Can there be a power basis for a totally real field of high degree?

A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of ...

**14**

votes

**1**answer

762 views

### Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.

**6**

votes

**0**answers

212 views

### Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...

**1**

vote

**0**answers

135 views

### The gcd of coprime polynomials evaluated at integers

Let $p(x),q(x)$ be coprime squarefree polynomials with
integer coefficients.
For integer $n$ is $\gcd(p(n),q(n))$ bounded by
an absolute constant?
In case the answer is negative what is
the fastest ...

**10**

votes

**5**answers

800 views

### Is $x^p-x+1$ always irreducible in $F_p[x]$?

It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...

**17**

votes

**1**answer

1k views

### How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...

**10**

votes

**3**answers

252 views

### Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$

It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of ...

**3**

votes

**1**answer

285 views

### Counting solutions modulo primes

Let $P(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ of degree $n.$ By $N(k,x)$ we denote the number of primes up to $x,$ such that $P(x)$ has exactly $k$ solutions in $\mathbb{Z}_p.$ Is it ...

**3**

votes

**0**answers

120 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...

**3**

votes

**0**answers

128 views

### Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...

**6**

votes

**2**answers

588 views

### Would such polynomial identity exist? (related to sum of four squares)

Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.
Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$?
I suppose the answer is negative.
If this is possible, ...

**1**

vote

**0**answers

80 views

### Irreducibility of reflexive compositions of polynomials (II)

Given an unknown polynomial $p \in \mathbb{Z}[x]$, write on the blackboard a finite expression that defines a polynomial $q \in \mathbb{Z}[x]$ by using only constants $1, x, p$ of the ring ...

**2**

votes

**0**answers

156 views

### square-free parts of values of polynomials

Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:
$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$
$$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...

**9**

votes

**3**answers

1k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...

**34**

votes

**2**answers

833 views

### A curious identity related to finite fields

To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$
of $q$ elements we associate the number $N(a_1,a_2,a_3)$
of elements $a_0\in \mathbb F_q$ such that the polynomial
...

**4**

votes

**2**answers

306 views

### Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of ...

**10**

votes

**1**answer

251 views

### Reducibility of polynomials maps

Motivated by this question.
Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ .
Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$.
If some $f^k(x)$ is reducible, the rest iterates will be ...

**1**

vote

**0**answers

53 views

### Irreducibility of reflexive compositions of polynomials

Let $m$ be a positive integer. For $k = 1, 2, 3, ... m$, fix $g_k(x_1, ..., x_{k + 1}) \in \mathbb{Z}[x_1, ..., x_{k + 1}]$.
For any polynomial $p(x) \in \mathbb{Z}[x]$, let
$P_0(x) = p(x), P_1(x) ...

**3**

votes

**2**answers

266 views

### Irreducible polynomials in $\mathbb{Q}_p((X))[Y]$

I'm looking for some criteria for the irreducibility of polynomials with coefficients in $\mathbb{Q}_p((X))$.
In particular, is the polynomial $Y^2+1$ irreducible over $\mathbb{Q}_3((X))$? And how ...

**7**

votes

**0**answers

79 views

### Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...

**1**

vote

**0**answers

218 views

### When $x^n - (x+a)^{n-k} (x+b)^k$ is irreducible?

Fix a nonzero integer $a$ and a positive integer $k$. I'm looking for some criterion to establish for which nonzero integers $b$ and $n \geq 2$ the polynomial
$$f(x) := x^n - (x+a)^{n-k} (x+b)^k$$
is ...

**2**

votes

**2**answers

227 views

### Irreducible Polynomials from a Reccurence

This question is inspired by a recent one : Let $c$ be a variable and define a sequence by $a_0=0$ $a_1=1$ and $a_{n+1}=a_{n}c-a_{n-1}$ . So
$$\begin{align*}
a_2 &= c
\\ a_3 &={c}^{2}-1= ...

**10**

votes

**3**answers

653 views

### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**0**

votes

**0**answers

83 views

### Lower-Upper bounds on the cardinality of a set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...

**3**

votes

**1**answer

264 views

### Menon’s identity

I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity
Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...

**4**

votes

**2**answers

307 views

### On $a^4+nb^4 = c^4+nd^4$ and Chebyshev polynomials

In a 1995 paper, Choudhry gave a table of solutions to the quartic Diophantine equation,
$a^4+nb^4 = c^4+nd^4\tag{1}$
for $n\leq101$. Seiji Tomita recently extended this to $n<1000$ and solved ...

**4**

votes

**3**answers

325 views

### Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...

**7**

votes

**3**answers

374 views

### How few terms may appear in a polynomial with given (cyclotomic) roots and nonnegative coefficients?

Given $W \subset \mathbb C$, let $S_W$ be the set of polynomials in $\mathbb R[x]$ that vanish on $W$ and have only nonnegative coefficients.
Warm-up question: It's clear that if $W$ contains a ...

**1**

vote

**0**answers

203 views

### Every antisymmetric (alternating) polynomial is divisible by Vandermonde product

I am looking for a proof of the statement:
Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result.
The statement is really ...

**1**

vote

**1**answer

147 views

### If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings):
If $F(a_1, \ldots, a_k)$ is a ...

**0**

votes

**1**answer

169 views

### Integral points on genus 0 curves related to polynomial identities

Consider the identity
$$(x+1)^5+(y+1)^5 = (2x + 6y) ( -x +y )^4 + f(x,y)$$
where $f(x,y)=-x^5 + 2*x^4*y + 12*x^3*y^2 - 28*x^2*y^3 + 22*x*y^4 - 5*y^5 + 5*x^4 + 5*y^4 + 10*x^3 + 10*y^3 + 10*x^2 + ...

**9**

votes

**3**answers

723 views

### Why is the gcd so large in an identity related to the $abc$ conjecture?

Consider the identity
$$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$
Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...

**28**

votes

**3**answers

2k views

### Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And ...

**5**

votes

**2**answers

150 views

### Collision polynomials

Consider $P_n(x)$ polynomials defined through the recurrence relations
$$P_n(x)=2(1-x)P_{n-1}(x)-(1+x)^2P_{n-2}(x),$$ with $P_0(x)=1$ and $P_1(x)=1-3x$.
In fact, the explicit solution of these ...

**10**

votes

**1**answer

390 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**33**

votes

**2**answers

2k views

### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...

**1**

vote

**2**answers

144 views

### Polynomials giving Lower Degree Elements in an Algebraic Number Field

My earlier related question
Lower Degree Elements in an Algebraic Number Field
has been given a clean answer for the first part. My present question is below:
Take a number field ...

**20**

votes

**4**answers

1k views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**18**

votes

**5**answers

657 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...

**1**

vote

**0**answers

277 views

### When integer polynomials take integer values, does their GCD also take integer values?

Suppose you have two polynomials $P$ and $Q$ with integer coefficients. Let their GCD (more specifically, the smallest integer multiple of the GCD in $\mathbb Q$ such that all the coefficients are ...

**4**

votes

**0**answers

262 views

### $a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from genus 0 curve ...

**7**

votes

**0**answers

220 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?
...

**9**

votes

**2**answers

595 views

### When polynomial f(x^2) can be factored as g(x)·g(-x) ?

In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where ...