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

learn more… | top users | synonyms

0
votes
1answer
39 views

Prove the following for integers

How can I show that ...
0
votes
0answers
18 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 w to the monomial symmetric ...
2
votes
2answers
39 views

Roots of simultaneous power sum equations (numerically or otherwise)

I'm a physicist, and I've come across a problem in my research where I need to solve a set of equations looking like (e.g. in 3D) $$r_1 + r_2 + r_3 = k_1$$ $$r_1^2 + r_2^2 + r_3^2 = k_2$$ $$r_1^3 + ...
1
vote
2answers
14 views

Implications of zero elemntary symmetric polynomials over a finite field

For a prime $q$ and an integer $n<q$, consider working over the finite field of $q^n$ elements. Denote by $s_n^k$ the $k$-th elementary symmetric polynomial in $n$ variables. That is, ...
0
votes
0answers
31 views

LCM of two polynmials when they are represented as point-value.

I`m wondering if we can obtain least common multiple of two polynomial when each polynomial represented as point-value. To be more clear, can we do any computation on these point-values and obtain ...
-3
votes
0answers
17 views

Polynomial derivative from evaluated points

If we evaluate a polynomial on some points, denoted by x1,x2,..xn, to obtain some value denoted by y1,y2,..yn. Can we obtain dth derivative of this polynomial by knowing only the values of y1,...yn ? ...
0
votes
0answers
25 views

Looking for general proof of a sum of an additive form of elementary symmetric polynomials

For sake of avoiding complicated general formulation I try to formulate in the special case of a set of 3 numbers $M=\{a_1,a_2,a_3\}$ with e.g. $a_i\in\mathbb R$. The sum I am looking for is in this ...
2
votes
1answer
79 views

How big are Kostka-Numbers

Let $n\in\mathbf{N}$ and $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ be integers such that $\sum_{i=1}^\ell\lambda_i=n$. To this partition consider the Schur-Polynomial $s_\lambda$. When expressed in ...
1
vote
1answer
46 views

How many terms are in $\sum \alpha_1^{a_1}\alpha_2^{a_2}\cdots \alpha_r^{a_r}\alpha_{r+1}\alpha_{r+2}\cdots \alpha_s$

Suppose that $\alpha_1, \cdots, \alpha_n$ be $n$ roots of the polynomial equation $p(x)=0$ of degree $n$. I was studying on symmetric polynomial and have come accross of several problems on like $\sum ...
1
vote
0answers
24 views

Necessary condition for symmetric sums

It is easy to see that if a function $f(x_1,x_2,x_3)$ can be written in the form: $$ f(x_1,x_2,x_3) = g(x_1,x_2) - g(x_1,x_3) + g(x_2,x_3) $$ for some function $g$, then we have: $$ f(x_1,x_2,x_3) - ...
0
votes
0answers
31 views

Perform $x_1^3+x_2^3+\cdots+x_n^3$ by basic symetric polynomials.

Perform $x_1^3+x_2^3+\cdots+x_n^3$ by basic symetric polynomials $\sigma_1,...,\sigma_n$. $\sigma_1=x_1+x_2+\cdots x_n\\ \sigma_2=\sum\limits_{i<j}x_i x_j\\ ...\\ \sigma_n= x_1.x_2...x_n$ I ...
4
votes
0answers
175 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
357 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
34 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
444 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
33 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
36 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
68 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
23 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
54 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
78 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
162 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
41 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
42 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
36 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
101 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
71 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
40 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
42 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
54 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
60 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
89 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
30 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
182 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
55 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
116 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
88 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
8answers
351 views

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

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
1
vote
1answer
92 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
1answer
64 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
90 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
229 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
38 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
102 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
52 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
78 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
127 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 $2n$ ...