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

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3
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0answers
36 views

Writing sum of square roots with symmetric polynomials

I want to write the function $$ F_N=\sum_{i=1}^N\sqrt{x_i} $$ in terms of the $N$ elementary symmetric polynomials of the $N$ positive variables $x_1,\dots,x_N$. The $N=1$ case is trivial, as we ...
3
votes
2answers
304 views

Sum of cubes of roots of a quartic equation

$x^4 - 5x^2 + 2x -1= 0$ What is the sum of cube of the roots of equation other than using substitution method? Is there any formula to find the sum of square of roots, sum of cube of roots, and sum ...
1
vote
0answers
28 views

Dimension of the Image of Young Projectors corresponding to Tensor factors.

Suppose I define the action of the symmetric group on abstract tensors as shuffling indices. I know this is very naive. I apologise, I am a physicist and working on a problem that involves tensors ...
2
votes
3answers
406 views

How to solve a system of three nonlinear equation in a simple way

Given the system: $$ \begin{cases} x^2y^2+x^2z^2=axyz & \\ y^2z^2+y^2x^2=bxyz &\\ z^2x^2+z^2y^2=cxyz \end{cases} $$ The solution could be gotten in a very tedious way. Is ...
1
vote
1answer
31 views

Basic Manipulation of Adams operations in R(G)

This is part of an exercise in Serre's representation theory book I am self-studying, but mostly it is about manipulation of symmetric polynomials. Let $\rho$ be a representation of a finite group ...
1
vote
1answer
24 views

natural isomorphism of polynomial functions on $V$ and $S(V^*)$

In Humphreys, reflection groups and coxeter group book, Humphreys denotes $S(V^*)$ as the ring of polynomial function on the finite dim vector space $V$. Why we are considering $S(V^*)$ rather than ...
4
votes
0answers
64 views

Combinatorics and symmetric functions

(The actual questions in this posting are at the bottom.) Occasionally someone asks here how to show that every nonempty finite set has just as many subsets of odd cardinality as of even cardinality ...
1
vote
1answer
21 views

Symmetric polynomials in some variables

Let $A_k$ be a set of polinomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ symmetric on $k<n$ variables: $$f(x_{\sigma(1)},\ldots, x_{\sigma(k)},x_{k+1}, ...
0
votes
1answer
44 views

Proving Newton's identities

Assume $F$ is a field of zero characteristic. Denote the elementary symmetric polynomials of $n$ variables by $e_k$, $\quad k=\overline{1,n}$. Let the symbol $\sum ax_1^{i_1}\dots x_n^{i_n}$ denote ...
1
vote
1answer
57 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
9
votes
1answer
152 views

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
0
votes
1answer
40 views

Using elementary polynomials to solve system of linear polynomials

Problem Statement I am given a finite set of monic polynomials in t, parameterized by $r_i$ $X_i = t - r_i$ where the $r_i$ are guaranteed unique. Neither $t$ nor $r_i$ are known, only $X_i$. I ...
3
votes
2answers
41 views

An elementary symmetric polynomial question

If we have three complex numbers $a,b,c$ such that the three elementary symmetric polynomials $a+b+c$, $ab+ac+bc$, and $abc$ are all integers, what characteristics can one deduce about $a,b,c$? For ...
2
votes
1answer
33 views

If $f$ is an anti-symmetric polynomial, then $f=g\prod_{1\leq i < j\leq n}(X_i-X_j)$ for some $g$ symmetric

So we have the situation that $f\in K[X_1,...,X_n]$ is anti-symmetric, which means that $\sigma (f)=\pm f$ where it is a plus if $\sigma$ is an even permutation on the $X_i$ and a minus if it is not ...
6
votes
0answers
95 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
2
votes
1answer
69 views

Equation on $\mathbb{R}$ : $(x+y+z)^3=x^3+y^3+z^3$

How would i find all the $(x,y,z)\in \mathbb{R}$ verifying $(x+y+z)^3=x^3+y^3+z^3$ ?
1
vote
0answers
39 views

Non-symmetric polynomials, game

This is a game I thought was easy but appears to be too hard for me... I'm trying to find a polynomial in x,y,z (they commute) such that permutations of the variables only give rise to 2 different ...
4
votes
2answers
41 views

Prove or disprove the system about $n$th power has only one solution $x=y=1$

$$\begin{cases}x^n+y^n=2\\x+y=2\end{cases}\;,\;n\in\mathbb{N}\;,\;x,y\in\mathbb{R}\;,\;n>2$$ I have tried to show that $\displaystyle y'=-\frac{x^{n-1}}{y^{n-1}}=-1$ $$......$$ therefore $x=y=1$ ...
0
votes
0answers
27 views

size of orbit of a polynomial under the action of $S_n$

Given a polynomial $\Phi$ in the $n$ variables $X_1,\ldots,X_n$, the values of $\Phi$ are defined to be the polynomials in the orbit of $\Phi$ under the action of the full symmetric group $S_n$. For ...
1
vote
2answers
52 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
1
vote
0answers
54 views

Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions

Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i < j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to ...
0
votes
0answers
44 views

Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
1
vote
3answers
81 views

Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
0
votes
0answers
26 views

Identity with symmetric rational functions

I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as ...
2
votes
3answers
170 views

How to solve the following pair of non-linear equations

$x^3+y^3=152\\x^2y+xy^2=120$ Best regards !
2
votes
2answers
54 views

Have I done something wrong in solving the following pair of equations?

Question given: Solve,$3x^2-5y^2-7=0\\3xy-4y^2-2=0$ What I have done so far: $$ ...
2
votes
4answers
113 views

How to solve the following system of equations.

$2x^2-3xy+2y^2=2\frac{3}{4}\\x^2-4xy+y^2+\frac{1}{2}=0$ I tried all the methods that I know, but I could't isolate $x$ or $y$ to form one equation.
2
votes
2answers
82 views

Uniqueness of a solution of the system of equations

A friend asked me the following question several days ago, and we still do not have a solution. Prove that the system of equations below has only the solution $(x, y, z)=(1, 1, 1)$. $$ \begin{cases} ...
5
votes
7answers
313 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
1
vote
1answer
83 views

Symmetric polynomial and vieta's formulas

Can you help me solve this problem: Express the following symmetric rational function $$\frac{x_1}{-6x_2^2 - 7x_3x_2 - 6x_3^2} + \frac{x_3}{-6x_1^2 - 7x_2x_1 - 6x_2^2} + ...
0
votes
0answers
51 views

Using Polya's Theorem to check positivity of a multivariate polynomial

I wish to check if a homogeneous polynomial of total degree 4 is positive definite. The polynomial is of the form $$P(u,v,x,y) = \sum_i\alpha_iu^{i_1}v^{i_2}x^{i_3}y^{i_4}$$ with $0 \le i_j \le 2$, ...
0
votes
2answers
87 views

Number of real solutions for the following set of equations? [closed]

How to solve the following set of equations for real values of $x,y$ and $z$? $$x^2-y^2=z$$ $$y^2-z^2=x$$ $$z^2-x^2=y$$ $(0,0,0)$ is an obvious one solution.
3
votes
1answer
141 views

Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$

How to prove the following equality? $$(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$$ I did it $$\begin{aligned} a^2b + a^2c + ab^2 + cb^2 + bc^2 + ac^2 + 2abc &=a^2(b + c) + bc(b + c) + a(b^2 + ...
0
votes
0answers
32 views

Elementary symmetric polynomials: Why is this sufficient?

Let $x_1,\dotsc,x_n$ be algebraically independent over $K$. $S_n$ acts on $K(x_1,\dotsc,x_n)$ by permuting the $x_i$s, giving rise to distinct field automorphisms of $K(x_1,\dotsc,x_n)$, i.e. a ...
4
votes
2answers
106 views

Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
1
vote
1answer
99 views

Discriminant is zero iff $f\in K[X]$ has repeated roots

I have to prove the statement in the title. Proving from right to left is easy. Now from left to right: $D=(\alpha_1-\alpha_2)^2(\alpha_1-\alpha_3)^2\cdots(\alpha_{n-1}-\alpha_n)^2$ where $\alpha_i$ ...
1
vote
0answers
45 views

Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
4
votes
3answers
76 views

Finding value of equation without solving for a quadratic equation

How do I go about solving this problem: If $α$ and $β$ are the roots of $x^2+2x-3=0$, without solving the equation, find the values of $α^6 +β^6$. In my thoughts: I commenced by expanding $(α ...
4
votes
0answers
120 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
1
vote
1answer
39 views

Can we represent a symmetric curve by a parameter with symmetry?

Question : Can we represent the following curve $C$ by one parameter $t$ as $x=f(t),y=g(t),z=h(t)$ with symmetry? The curve $C$ in the $xyz$ space is defined as $$\begin{cases} x^2+y^2+z^2=1 ...
0
votes
3answers
120 views

Find all $x,y,z$ satisfying $xy=z-x-y$ and cyclic permutations

Find all ordered pairs $(x,y,z)$ real numbers, which satisfy the following system of equations: $$xy=z-x-y\\xz=y-x-z\\yz=x-y-z$$
0
votes
3answers
142 views

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?
1
vote
2answers
48 views

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ …

Problem : If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ then the value of $(1-\alpha_1)(1-\alpha_2)(1-\alpha_3)(1-\alpha_4)$ is (a) 2$\sqrt{3}$ ...
1
vote
0answers
103 views

Symmetric function theorem and Galois Theory — How deep is the connection?

By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric ...
5
votes
1answer
82 views

Taking a power of a polynomial to make it symmetric

Suppose I have a non-symmetric multi-variable polynomial in $n$ variables $P(x_1,x_2,...,x_n)$. For example $P$ might be $x_1^2+x_2$ or $x_1-x_2$ Under what conditions will some power $m$ of $P$ ...
1
vote
0answers
71 views

How to solve this system equation of polynomials?

I have: $F(x) + G(x) = 1 + F(x)*M,$ $G(x) = T_{1}(x) + T_{2}(x) + ... + T_{N}(x)$ $F(x )x^{a_{i}} = T_{i}(x) \times C_{i}(x) + \sum_{j \leq N} T_{j}(x) \times P_{ji}(x)$ In which $M, N, a_{i}, ...
11
votes
1answer
158 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
0
votes
1answer
275 views

Explain the terms : homogeneous , symmetric , anti-symmetric , cyclic with respect to polynomials.

In answers to some of my previous questions , a lot of people used the terms homogeneous polynomial ( in a,b,c ) (under permutations of variables ) , cyclic polynomial ( in a,b,c) (under permutations ...
0
votes
1answer
52 views

Factoring the group action on $\prod X_k^k $ gives another group?

Let's say you have a monomial symmetric polynomial, like the following $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 +X_2^1X_1^2X_3^3X_4^4 + \text{all permutations...} $$ Then you can ...
7
votes
0answers
189 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots ...