**20**

votes

**1**answer

700 views

### $f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible (over $\mathbb{Z}[x]$)?

**2**

votes

**0**answers

135 views

### Irreducibility of $x^m-g(y)$

Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?

**4**

votes

**0**answers

126 views

### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...

**1**

vote

**0**answers

188 views

### Continuous dependence of the roots of a polynomial on its coefficients

In their article "The roots of a polynomial vary continuously as a function of the coefficients" Gary Harris and Clyde Martin give a topological proof of the well-known theorem that the roots of a ...

**7**

votes

**1**answer

391 views

### Integer valued polynomial through some points with rational coordinates

I asked this question on MSE about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO.
If we have a set of points $(x_i,y_i)$ ...

**7**

votes

**2**answers

445 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...

**5**

votes

**1**answer

240 views

### Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...

**1**

vote

**0**answers

114 views

### First Description of how to Remove Radicals from Equations

Who first described the technique of removing radicals as indicated in the answers to questions Tools for Removing Radicals from Equations and Rewrite sum of radicals equation as polynomial equation ? ...

**3**

votes

**0**answers

318 views

### A question on partial fraction decompositions

This question concerns the mapping from the poles of a rational function to the coefficients of its partial fraction decomposition. In general, this mapping is not injective. I want to identify some ...

**1**

vote

**0**answers

324 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 = ...

**1**

vote

**0**answers

54 views

### Decomposition of polynomials with three variables

We use $\bigtriangleup _i$ to denote either multiplication or addition.
Suppose we have a polynomial $P(x,y,z)$ over some algebraic closed field such that:
There are $Q(x), W_1(x,y),W_2(x,z)$ ...

**14**

votes

**6**answers

2k 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 ...

**3**

votes

**1**answer

121 views

### Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows:
Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$
such that ...

**8**

votes

**1**answer

433 views

### Gaps between roots of trigonometric polynomials

[Cross-posted from Math.SE because I got no responses there.]
Given a polynomial in $e^{\mathrm{i}k t}$ of the form
$$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$
with $\bar c_{-k} = c_k$, ...

**9**

votes

**2**answers

533 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 ...

**15**

votes

**1**answer

865 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.

**2**

votes

**0**answers

107 views

### Existence of roots of high order polynomial over finite fields

I want to solve the following question:
Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...

**12**

votes

**3**answers

455 views

### Existence of solutions of a polynomial system

Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional real vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} ...

**4**

votes

**0**answers

230 views

### Polynomial dynamical systems

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...

**0**

votes

**1**answer

70 views

### Fitting a quadratic using regression when the y-intercept needs to be 0 [closed]

I'm trying to fit a quadratic $a_0 + a_1x + a_2x^2$ by Polynomial Regression:
$$
\begin{pmatrix}
n & \Sigma x_i & \Sigma x_i\\
\Sigma x_i & \Sigma x_i^2 & \Sigma x_i^3\\
\Sigma ...

**1**

vote

**2**answers

111 views

### Estimate maximal coefficient of a polynomial from a circle containing all roots

Suppose I have a polynomial
$$
p(x)=\sum_{i=0}^n p_ix^i.
$$
For simplicity furthermore assume $p_n=1$.
As it is well known we may use Gershgorin circles to give an upper bound for the absolute ...

**6**

votes

**0**answers

233 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 ...

**2**

votes

**0**answers

213 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**9**

votes

**1**answer

296 views

### Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$

Theorem. The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root.
Question. What ...

**1**

vote

**1**answer

72 views

### seeking reference on a theorem about sufficient conditions for an entire function with real coefficients to have only real zeros

I am seeking reference(s) on the following theorem about sufficient conditions for an entire function with real coefficients to have only real zeros.
Theorem:
Let $f_n(z)=\sum_0^n a_m z^m$ (with ...

**1**

vote

**0**answers

36 views

### Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials):
(a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$;
(b) the first $k$ ...

**3**

votes

**2**answers

289 views

### irreducible polynomials on the polynomial sequence

I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution ...

**6**

votes

**0**answers

172 views

### Find a symmetric polynomial with a projection divisible by a known polynomial

Consider the polynomial $Q$, a homogeneous quartic in seven variables:
$$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\
...

**1**

vote

**0**answers

214 views

### R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module.
Would $R$ noetherian ...

**4**

votes

**0**answers

45 views

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial
is a bivariate polynomial with positive integer coefficient which is a graph
invariant and can be defined recursively.
Evaluating it is $\#P$-complete even when restricted to ...

**4**

votes

**0**answers

155 views

### The equation $f(x)=f(a^n)^k$ always has a solution in $\mathbb{Q}$ [closed]

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ ...

**-1**

votes

**1**answer

124 views

### derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has
...

**4**

votes

**0**answers

121 views

### The closures in $C^0(\mathbb C,\mathbb C)$ of the set of integer valued polynomials

This question is closely related to the thread The closures in $C^0(\mathbb R,\mathbb R)$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients. (Recall that a ...

**3**

votes

**1**answer

213 views

### Polynomial without roots in a ring

Let $A$ be a commutative ring which is not an integral domain. I try to find a polynomial $P$ of $A[X]$ such that $d°P = 1$ and $P$ admits no root in any ring $B$ such that $A$ is a subring of $B$.

**5**

votes

**1**answer

223 views

### Literature about metapolynomials

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form
$$f(x_1,\cdots , x_k ...

**3**

votes

**2**answers

157 views

### Unimodular triangulation and Ehrhart polynomials

Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.
I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties
is integrally ...

**1**

vote

**0**answers

109 views

### polynomial with rational coefficients [closed]

Do you think that the following statement is true? Do you have any idea about the proof?
Let $\; f(x) \in \mathbb{C}[x]$ be a polynomial. If $f(n) \in \mathbb{Z}$ for an infinite number of $n \in ...

**3**

votes

**0**answers

111 views

### Discriminant polynomial generalizing the usual discriminant

I wonder if anybody has seen the following natural polynomial.
Given a monic univariate polynomial $P(z)$ of degree $N$, denote its roots by
$z_1,..., z_N$. Now form a new polynomial $Q(z)$ of ...

**2**

votes

**1**answer

108 views

### Roots of modified polynomials

Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...

**17**

votes

**2**answers

597 views

### Number of zeros of a polynomial in the unit disk

Suppose $m$ and $n$ are two nonnegative integers. What is the number of zeros of the polynomial $(1+z)^{m+n}-z^n$ in the unit ball $|z|<1$?
Some calculations for small values of $m$ and $n$ ...

**6**

votes

**7**answers

723 views

### Source for roots of matrix polynomials?

A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...

**4**

votes

**1**answer

230 views

### Stronger versions of Schwartz-Zippel for random linear subspaces

This is a (self-contained) followup question to http://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...

**2**

votes

**1**answer

160 views

### Lagrange Interpolation and integer polynomials

Suppose that there is a polynomial $P$ with integer coefficients such that $P(x_i)=y_i$ for $i=1,\ldots,n$. Is it true that the result of Lagrange interpolation through the data $(x_i,y_i)$ is a ...

**1**

vote

**0**answers

83 views

### Generalization of the Hermite-Biehler-Kakeya Theorem (2)

This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments.
Firstly we remark that: $f(x)+g(x)\cdot w$ is ...

**1**

vote

**0**answers

160 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 ...

**1**

vote

**0**answers

38 views

### Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...

**2**

votes

**2**answers

206 views

### Sextic resolvent has no rational root

An irreducible quintic $f(x)\in\mathbb{Q}[x]$, is solvable by radicals if and only if its sextic resolvent $\theta_f (y)=(y^3+py^2+qy+r)^2-2^{10}\Delta(f)y$ has a rational root ($\Delta$ is the ...

**2**

votes

**0**answers

268 views

### Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for ...

**5**

votes

**0**answers

171 views

### Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...

**3**

votes

**2**answers

356 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...