# Tagged Questions

**16**

votes

**1**answer

803 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 ...

**7**

votes

**3**answers

213 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 ...

**1**

vote

**0**answers

185 views

### A question about the Sylvester determinant

I am writing an invited article related to the diophantine equations like the abc-conjecture by the way of the determinant of the Sylvester matrix. We know that Sylvester never really finalize some ...

**3**

votes

**1**answer

254 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

113 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 ...

**2**

votes

**0**answers

124 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 ...

**5**

votes

**2**answers

537 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

75 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

147 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

768 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

282 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 ...

**7**

votes

**1**answer

236 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

52 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

262 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 ...

**6**

votes

**0**answers

77 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

215 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

213 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

588 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

79 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

220 views

### Menonâ€™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 ...

**5**

votes

**2**answers

296 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

308 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

355 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

153 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

144 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

156 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

698 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 + ...

**27**

votes

**3**answers

1k 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

141 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 ...

**8**

votes

**1**answer

333 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 ...

**30**

votes

**2**answers

1k 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

141 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 ...

**18**

votes

**3**answers

936 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

613 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

263 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 ...

**3**

votes

**0**answers

245 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

215 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

584 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 ...

**1**

vote

**2**answers

179 views

### Transformation of a bivariate polynomial into a homogeneous one

For a given a bivariate polynomial $P(x,y)$ with rational coefficients:
Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...

**6**

votes

**0**answers

333 views

### How many values a polynomial map misses?

Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...

**3**

votes

**1**answer

128 views

### Decision problem wrt Pairs of Polynomials with Integer Coefficients

Given two arbitrary polynomials $G(x)$ and $H(y)$, with integer coefficents, are there any circumstances in which it is possible to decide whether or not $G(x) = H(y)$ has solutions with $x, y \in ...

**11**

votes

**0**answers

496 views

### Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims
Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004
Is it really so?
As far as I know, it is open problem ...

**0**

votes

**1**answer

272 views

### Polynomial version of the conjecture about Power free-values of polynomials

The conjecture about Power free-values of polynomials
is:
Let $F(X)$ be a polynomial with integer coefficients and no repeated roots. For any $\epsilon > 0$, there exists a constant ...

**5**

votes

**1**answer

1k views

### Forbidden polynomial identities by the abc conjecture

The Masonâ€“Stothers theorem states
Let $a(t), b(t)$, and $c(t)$ be relatively prime polynomials such that $a + b = c$, with coefficients that are either real numbers or complex numbers. Then ...

**0**

votes

**1**answer

262 views

### Cyclotomic polynomial question [closed]

To avoid case distinction overload, I also call (say) $Z^3-Z^4$ cyclotomic. Just divide out the $Z=0$ solutions in the following if they offend.
In the following, all exponents are assumed to be ...

**0**

votes

**2**answers

162 views

### power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...

**5**

votes

**3**answers

844 views

### Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?

My qeustion is that,
is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...

**3**

votes

**1**answer

470 views

### When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...

**17**

votes

**2**answers

788 views

### Cyclotomic polynomials with coefficients $0,\pm1$

Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki ...