**17**

votes

**2**answers

248 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...

**7**

votes

**0**answers

118 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...

**4**

votes

**0**answers

38 views

### $G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$.
My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

**2**

votes

**0**answers

73 views

### $\mathbb{C}[x_1, \dots, x_n]$ is a free $\mathbb{C}[x_1, \dots, x_n]^{S_n}$-module with certain generators [duplicate]

Let the symmetric group $S_n$ act on $\mathbb{R}^n$ by permutation of coordinates. This makes $S_n$ a subgroup of $\text{GL}_n(\mathbb{R}$ and the algebra $\mathbb{C}[x_1, \dots, x_n]^{S_n}$ is the ...

**-5**

votes

**0**answers

103 views

### New way to solve Quadratic Equations [on hold]

A couple months ago, I was challenged to discover a new way to solve quadratic equations with one rule: I must think of it myself. At first I was hopeless, trying various sorts of random things, but I ...

**11**

votes

**2**answers

420 views

### No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$

See David Speyer's answer here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, ...

**1**

vote

**0**answers

31 views

### Connection between the Chebyshev polynomials and the Faber polynomials

From a comment on this question:
@draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), which 'invert" the cyclic partition ...

**2**

votes

**0**answers

35 views

### How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...

**0**

votes

**2**answers

156 views

### Is there a way to find out how many distinct roots a polynomial has? [closed]

Let say we have an arbitrary polynomial over the reals, and we do not know whether it is separable or not. Is there some algorithm to find the number of roots it has in the complex number?

**9**

votes

**1**answer

361 views

### $\pi e$ and an unfamiliar polynomial

Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral
$$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, ...

**29**

votes

**2**answers

919 views

### Polynomial $g:\mathbb R^n \rightarrow\mathbb R^n$ with no critical point may have no root

Version 1 (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n ...

**3**

votes

**0**answers

60 views

### Is there a natural covariant of sextic polynomials with the following coefficients?

Let
$$\displaystyle f(x) = a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 $$
be an irreducible sextic polynomial with integer coefficients. Write $\theta_1, \cdots, \theta_6$ for the ...

**3**

votes

**0**answers

135 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**3**

votes

**1**answer

285 views

### If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...

**6**

votes

**2**answers

371 views

### Seeking an explanation for a peculiar factorization

Recently during my work, I encountered the following family of sextic polynomials
$$\displaystyle x^6 - 3x^5 + cx^4 + (5 - 2c)x^3 + cx^2 - 3x + 1.$$
By plugging in various values of $c$, I noticed ...

**3**

votes

**2**answers

118 views

### Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:
Let $k=\mathbb{C}((t))$ and let ...

**16**

votes

**0**answers

247 views

### Zero curves of Tutte Polynomials?

There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...

**6**

votes

**0**answers

100 views

### About the properities of sum of powers of items in a polynomial

Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials:
\begin{eqnarray*}
...

**7**

votes

**0**answers

107 views

### Is $F_{f, c, \ell}$ a $G$-harmonic polynomial?

Let $G \subset \text{GL}_n(\mathbb{C})$ be a finite subgroup. The group $G$ acts naturally on $\mathbb{C}^1[\mathbb{C}^n]$ the space of degree $1$ homogeneous polynomials in $x_1, \dots, x_n$, i..e, ...

**0**

votes

**0**answers

85 views

### How to find solutions for four polynomial equations with four unknown variables using Resultant Theory

Can I use resultant theory (or polynomial resultant method) to find solutions for four simultaneous polynomial equations with four unknown variables?
So far, I could only find examples which uses two ...

**0**

votes

**0**answers

65 views

### Quotient modules of polynomial rings by maximal one sided ideal

Let R[X] be a ring of polynomials over an associative unital ring R which is not necessarily commutative. Let M be a maximal left ideal in R[X]. It is easy to see that if the intersection of M with ...

**1**

vote

**0**answers

17 views

### Complexity of finding algebraic dependency of polynomials over the rationals or in a finite field?

Let $f_1,\ldots f_m \in K[x_1,\ldots,x_n]$ where $K$ is $\mathbb{Q}$ or a finite field.
Q1 What is the complexity of finding all algebraic dependencies between $f_i$?
Q2 What is the ...

**11**

votes

**0**answers

109 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...

**14**

votes

**3**answers

672 views

### Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?

Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...

**3**

votes

**1**answer

64 views

### How to deduce the recursive derivative formula of B-spline basis?

Description
Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denotes a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$.
and the $i$-th B-spline basis function of ...

**2**

votes

**0**answers

44 views

### Hermite interpolation

I need a help to my problem, I would be grateful if anyone could help.
Let $\epsilon \in [0,1]$ and for an integer $n$ we consider a set of nodes $T_n={t_0,t_1,....t_n}$.
We define the function ...

**1**

vote

**2**answers

209 views

### When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?

This is a follow up on a previous question of mine.
Out of curiosity, I am wondering more generally when a closed form exists for
$$\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$$
where $P$ and $Q$ are ...

**2**

votes

**2**answers

113 views

### Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$

Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for ...

**1**

vote

**2**answers

134 views

### Generic polynomial for alternating group ${A}_{4}$ is not correct

I was validating the percentage of cases where the generic two parameter polynomial for Galois group ${A}_{4}$ is valid. We have
\begin{equation*}
{f}^{{A}_{4}} \left({x, \alpha, \beta}\right) = ...

**4**

votes

**0**answers

129 views

### A strange polynomial equality

In my answer to this question, I have obtained that the polynomial $p(x)$ of degree $2n$ with nonnegative values on $[-1,1]$ with $p(\pm1)=1$ has $\int_{-1}^1 p(x)\,dx\geq \frac{4}{(n+1)(n+2)}$, and ...

**1**

vote

**1**answer

71 views

### Showing that a particular area is small

Note: I posted this on math.stackexchange.com earlier (original post here: http://math.stackexchange.com/questions/1471331/showing-that-a-particular-area-is-small), but it received no responses and ...

**18**

votes

**1**answer

1k views

### Are the following identities well known?

$$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
...

**17**

votes

**2**answers

384 views

### Non-negative polynomials on $[0,1]$ with small integral

Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a_n = \min_{p\in P_n} \int_0^1 ...

**4**

votes

**0**answers

104 views

### Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...

**2**

votes

**1**answer

88 views

### Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary ...

**1**

vote

**1**answer

81 views

### Existence of real solutions for a system of linear and quadratic equations

Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not ...

**23**

votes

**2**answers

492 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**8**

votes

**1**answer

180 views

### Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...

**9**

votes

**1**answer

118 views

### a generalization of gamma matrices

Is it possible to find matrix solutions to the following :
$$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$
where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) ...

**1**

vote

**0**answers

35 views

### Minimum degree of a nonnegative polynomial uniformly approximating two constant values on two disjoint closed intervals

This is a one-dimensional problem over $\mathbb{R}$. Given $y_0, y_1 \ge 0$ with $y_0 \neq y_1$, and closed intervals $I_0$ and $I_1$ with $I_0 \cap I_1 = \emptyset$, define a partial function $f(x) = ...

**10**

votes

**3**answers

483 views

### A characteristic 2 polynomial recursion

Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ be defined by the recursion $$c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions
$$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad ...

**8**

votes

**0**answers

110 views

### Order of zeros for sparse polynomials mod $p$

It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c
\neq 0$, $f$ has a zero of order at ...

**5**

votes

**0**answers

214 views

### Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...

**0**

votes

**0**answers

51 views

### A question on boundary set

Suppose:
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, ...

**12**

votes

**0**answers

281 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

**1**answer

182 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

**0**answers

116 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

**2**answers

542 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 ...

**5**

votes

**1**answer

220 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

**2**answers

391 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 $\xi$ ...