Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

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3
votes
1answer
41 views

Almost symmetric polynomial?

Let's say we have a polynomial $(x-y)(y-z)(x-z)$. This is not a symmetric polynomial, but it almost is. Every permutation of the variables results in a polynomial whose factors are multiples of the ...
1
vote
1answer
34 views

Systems of equation

Find non-negative solutions of systems of equations: $$\begin{cases} x^2y^2+1=x^2+xy \\ y^2z^2+1=y^2+yz \\ z^2x^2+1=z^2+zx \end{cases} $$ My work so far: 1) $(1;1;1) - $ solution. 2) ...
1
vote
1answer
97 views

Solve this systems to condition $3x^3(x+1)^2=2y^2(z+3)^3$

if $x,y,z$be postive real numbers, solve systems of this following equation $$ 3x^3(x+1)^2=2y^2(z+3)^3\tag{1}$$ $$3y^3(y+2)^2=2z^2(x+1)^3\tag{2}$$ $$3z^3(z+3)^2=2x^2(y+2)^3\tag{3}$$ My approach is ...
1
vote
0answers
23 views

System of Nonlinear Equations (sum of powers)

I want to show the only solution to the following system of equations is the trivial one ($x_{i} = 0$). I don't know if this is true, but I think it should be. Let $x_{i} \in \mathbb{C}$ for $1 \le i ...
4
votes
1answer
50 views

Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
0
votes
1answer
14 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
2
votes
1answer
41 views

nth power symmetric polynomial in terms of Schurs polynomial

The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
0
votes
0answers
13 views

Schur polynomials with variables raised to a fixed power

For a partition $\lambda$ let $s_{\lambda}(x_1,\dots,x_k)$ denote the Schur polynomial in $k$ variables associated to $\lambda$ (let's assume that $k$ is sufficiently large compared to $\lambda$ that ...
9
votes
1answer
150 views

Geometry of Elementary Symmetric Polynomials

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ...
1
vote
1answer
37 views

A binomial symmetric sum

Denote \begin{align*} \text{Sym}_k(\textbf{x})=\sum_{i_1<\cdots<i_k}x_{i_1}\cdots x_{i_k} \end{align*} as the $k$th elementary symmetric sum in monomials $\textbf{x} = (x_1, \cdots, x_n)$. If ...
0
votes
0answers
26 views

Help with deriving Newton's Identities

I am trying to derive Newton's identities for symmetric sums, namely in the case where $k > n \geq 1$ \begin{equation} \sum_{i=k - n}^k (-1)^{i + 1}S_{k-i}P_i = 0 \end{equation} where $S_k$ is the ...
1
vote
1answer
50 views

What are my polynomials called? (and where can I read about them)?

I've run across some polynomials that are natural enough I imagine must have been named and studied, but don't know what they're called. In the ring $\mathbb{Z}[x_i,y_i : i \in I]$ one has ...
0
votes
1answer
16 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
1
vote
0answers
31 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
0
votes
2answers
56 views

Compute the trace of $\text{Sym}^2 \left(f \right)$ and that of $\text{Sym}^3 \left(f \right)$.

Consider the linear map $f: \mathbb{C}^3 \to \mathbb{C}^3$ defined by the matrix $$\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}.$$ Compute the trace of ...
3
votes
2answers
46 views

The Fundamental Theorem of Symmetric Polynomials

This theorem stays that any symmetric polynomial can be expressed as a polynomial of elementary polynomials. So let's suppose I have a polynomial $f(x_1,x_2,...,x_n)$ in $R[\mathbb{X}]$. I can find a ...
1
vote
1answer
66 views

Solve the system of equations $\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$

Find the number of real solutions to the system of equations $$\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$$ Let $f(x) = x^3-3x$ then for $x\in \mathbb{R}-(-2,2)$ we have $x_1 ...
3
votes
3answers
61 views

Solving system of three quadratic equations

$$\begin{cases} x^2 = yz + 1 \\ y^2 = xz + 2 \\ z^2 = xy + 4 \end{cases} $$ How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got ...
3
votes
1answer
73 views

Is it possible to “depress” any term in a polynomial with a suitable substitution?

If we have a degree $n$ polynomial $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$$with coefficients in a field, say $\Bbb C$, for concreteness, it is well known that the substitution $y= x + ...
1
vote
4answers
113 views

Factorize $(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$

I am unable to factorize this over $\mathbb{Z}:$ $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$$ Since, this from an exercise of a book (E. J. Barbeau, polynomials) it must have a neat ...
2
votes
1answer
42 views

if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
0
votes
1answer
36 views

Efficient way to compute the symmetric reduction of special polynomials (specially for resolvents)

By the Fundamental Theory of Symmetric Polynomials every symmetric polynomial in $K[x_1, \dots, x_n]$ can be written uniquely in the elementary symmetric functions $s_1, \dots, s_n$. I know there are ...
0
votes
0answers
16 views

Is there any Newton type identity for the following ordered matrix symmetric polynomial?

I have the following sum of matrices $A_i,\ i=1,2,\cdots,\ n$ $$\sum_{n\ge i_1>i_2>\cdots>i_k\ge 1}A_{i_1}A_{i_2}\cdots A_{i_k}$$ This looks like an elementary symmetric polynomial but it is ...
0
votes
0answers
48 views

Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by ...
2
votes
1answer
27 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
2
votes
4answers
56 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
0
votes
0answers
14 views

Concave property on elementary symmetric polynomials

Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}} $ ...
1
vote
0answers
55 views

Center of the group algebra of the symmetric group

How to prove that the center of the group algebra of the symmetric group is generated by 1-cycle conjugacy classes? I mean, that the center (consisting on class functions) is multiplicatively ...
0
votes
2answers
65 views

Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$. What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different? Edit: How can I show that ...
-1
votes
1answer
72 views
0
votes
1answer
45 views

prove a polynomial identity..

The equation is that $h_m(x_1, \cdots, x_n, a)-h_m(x_1, \cdots, x_n, b)=(a-b)h_{m-1}(x_1, \cdots, x_n, a, b)$ where $h_m$ is a complete homogeneous symmetric polynomial. See and find several ...
3
votes
4answers
84 views

Solving Symmetrical Equations Algebraically

I'm doing some Cambridge STEP papers and have come across a tricky set of equations. \begin{align*} 99 &= c^3 + 6 cd^2 \tag{1} \\ 70 &= 3c^2d + 2d^3 \tag{2} \end{align*} From looking ...
0
votes
1answer
44 views

Number of solutions for system of elementary symmetric functions?

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ...
1
vote
2answers
57 views

How to solve system: x_1+x_2+…+x_n=a [closed]

How to solve this system: $$\left\{\begin{matrix} x_1 & + &x_2 & + & \ldots & + & x_n &= & a \\ x^2_1& + &x^2_2 &+ & \ldots & + &x^2_n ...
1
vote
0answers
53 views

An Elementary Solution to a Polynomial Problem?

The following problem is from Larson's problem solving through problems: If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ ...
0
votes
1answer
120 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
4
votes
1answer
86 views

Properties of the 'forgotten' symmetric polynomials

In I.G. Mcdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions $f$ are introduced very briefly as the result of applying an involution $\omega$ to the monomial ...
1
vote
4answers
53 views

Homework: Sum of the cubed roots of polynomial

Given $7X^4-14X^3-7X+2 = f\in R[X]$, find the sum of the cubed roots. Let $x_1, x_2, x_3, x_4\in R$ be the roots. Then the polynomial $X^4-2X^3-X+ 2/7$ would have the same roots. If we write the ...
4
votes
0answers
30 views

Arnold's combinatorial description of entropy.

V.I. Arnold says that entropy is related to the asymptotic behaviour of polynomial coefficients. This is mentioned in his book "Dynamics, Statistics and Projective Geometry of Galois Fields". Here ...
3
votes
1answer
62 views

Symmetry planes in spherical harmonic basis

Suppose I have a function $f(x):S^2\rightarrow\mathbb{C}$ in the degree four spherical harmonic basis: $$f(\theta,\varphi):=\sum_{k=-4}^4a_kY_4^k(\theta,\varphi).$$ I have two related questions: Is ...
2
votes
1answer
86 views

How to solve this set of symmetric polynomial expressions

So there's this set of polynomial expressions with degree n=3: $$ \left\{ \begin{array}{c} x_1 + x_2 + x_3 = a \\ x_1^2 + x_2^2 + x_3^2 = b \\ x_1^3 + x_2^3 + x_3^3 = c \end{array} \right. $$ How to ...
1
vote
1answer
52 views

System of three equations with lots of symmetry and 6 unexpected (?) solutions.

I'm interested in the system of equations: $a(b^2+c)=c(c+ab)$ $b(c^2+a)=a(a+bc)$ $c(a^2+b)=b(b+ac)$ It is easy to see that $a=b=c=t$ are solutions for all $t$, in fact these are the only real ...
2
votes
1answer
64 views

Representation of eigenvector product using matrix elements

Let $A$ be a $n \times n$ real matrix, $(\lambda_i, v_i)$ be the $i$-th (eigenvalue, eigenvector) of $A^T$, and $x(t)$ be a vector of $n$ functions $x_i(t)$. For $\frac{d x(t)}{dt}=A x(t)$, the ...
4
votes
1answer
817 views

Character of the $n^{\text{th}}$ symmetric power of the standard representation of $S_3$

So I am working out Fulton-Harris's Representation Theory text. For $S_3$, there is the standard representation $V$ which is two dimensional. That's all great and fine. Let $Sym^k(V)$ be the symmetric ...
8
votes
2answers
343 views

Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?

User Eric Gregor and I were talking in chat and he mentioned this question and postulated the possibility of an approach through symmetric polynomials. After some thinking, I came to this: ...
1
vote
2answers
24 views

Possible values for this specific line of variables.

I have this line of numbers: xy + z = xz + y = yz + x I need to find out all the possible values of x, y and z in this equation. Thank you!:) My usual problem ...
0
votes
1answer
46 views

$(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials?

Is it possible to express $(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials ? What I tried was ...
6
votes
1answer
1k views

Proofs of The Fundamental Theorem of Symmetric Polynomials

I have been considering a few proofs of this theorem, and I noticed that a few of them (for example Proof 1, and Proof 2) prove the theorem first for homogeneous symmetric polynomials and then ...
1
vote
1answer
26 views

polynomials in terms of elementary symmetric polynomials

Let a polynomial of $2n$-variables be $$ f(x_1,\cdots,x_n,y_1,\cdots,y_n)=\prod_{i,j=1}^n(1+x_i+y_j). $$ Let the elementary symmetric polynomials be $\alpha_1=\sum_{i=1}^n x_i$, ...
2
votes
0answers
22 views

terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $ e_1,\cdots,e_n$ and $$ \mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n]. $$ We define a ...