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

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2
votes
1answer
34 views

Find the equation which has key root $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$

In my last question which was Proving $x=\sqrt{a}+\sqrt{b}$ is the key root to solve $x^4-2(a+b)x^2+(a-b)^2=0$ ,I could find the coefficients(were very easy) of fourth-degree equation, so I went to ...
4
votes
1answer
60 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
0
votes
1answer
42 views

Solving a simple systems of equations

Update: 1) As @Amzoti mentioned, I made a mistake in the mathematica code. There should be spaces between x, y and z. So now the following code works: ...
5
votes
4answers
331 views

Ring of polynomials as a module over symmetric polynomials

Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free ...
0
votes
1answer
77 views

How can I convince students a certain polynomial equation is symmetric?

How can I convince students that $p(x)=0$ is a symmetric equation if they ask me, where $p(x)$ is polynomial of degree $n$ with reals coefficients. For example : $A(x)=2x^4-9x^3+8x^2-9x+2=0 $ is ...
3
votes
2answers
27 views

Prove coefficients of polynomial are elementary symmetric polynomials

I want to show that for the $k$-th elementary symmetric polynomial $s_k:=\sum_{i_1\lt\cdots\lt i_k}X_{i_1}\cdots X_{i_k}\in R[X_1,\ldots,X_n]$ a monic polynomial that factors $\prod_{i=1}^n ...
3
votes
0answers
32 views

Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over ...
0
votes
0answers
20 views

Finding a generalized form for taking the n$^{th}$ derivative of a falling factorial

I would like to make $$ \frac{d^n}{dx^n}[(x)_c] = n! \times e_{c-n}(x,x-1,\cdots,x-c+1) $$ Where $e_{c-n}(x,x-1,x-2,\cdots,x-c+1)$ is the elementary symmetric polynomial function But lets say that ...
1
vote
1answer
66 views

Irreducibility over $\mathbb{C}$ of symmetric polynomials

Problem. Find all elementary symmetric polynomials that are irreducible over $\mathbb{C}$. My attempt. It's easy to see that if we have polynomial $f(x_1, \dots, x_n)$ and it can be reduced to ...
2
votes
0answers
28 views

multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
2
votes
1answer
39 views

Write the determinant as a polynomial expression in the elementary symmetric polynomials

How to write $\det\begin{bmatrix}x_1&x_2&x_3&x_4\\x_2&x_3&x_4&x_1\\x_3&x_4&x_1&x_2\\x_4&x_1&x_2&x_3 \end{bmatrix}$in terms of elementary symmetric ...
1
vote
1answer
29 views

Representing a symmetric function in elementary symmetric functions

I'm trying to represent the following in the elementary symmetric functions base: $ \sum\limits_{i \neq j} x^2_i x_j $ and $ \sum\limits_{i \neq j} x_i^2 x_j^2 $ I don't really know how to ...
0
votes
0answers
17 views

Convert newton's identities into a explicit function?

Is it possible to create a general explicit function (independent and dependent variable) of Newton's Identities? I apologize in advance if this question is somewhat vague. I will try to clarify if ...
0
votes
1answer
35 views

Build a polynomial

I have $f=x^3 + ax^2 +bx +c \in \mathbb C[x], \alpha_1,\alpha_2,\alpha_3 \in \mathbb C$ are roots of $f$. $\beta_1 = {\alpha_1 \over \alpha_2} + {\alpha_2 \over \alpha_3} + {\alpha_3 \over \alpha_1}, ...
1
vote
1answer
21 views

Polynom as sum/product of symmetric polynoms

I have a polynom $(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1)$ and I need to express as sum/product of elemental symmetric polynoms $s_1,s_2,s_3$. I know there is an algoritm for ...
0
votes
1answer
31 views

A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$ \frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1) $$ My first question is, how can I prove this? It seems trivial ...
4
votes
0answers
41 views

Question about primes of polynomial type.

It is well known that $50$ % of the primes are of the form $x^2 + y^2$. Many variants exists where a rational amount of primes is of some integer polynomial form. But I wonder ; are there integer ...
6
votes
2answers
59 views

$a_1^k+a_2^k+\ldots+a_n^k$ integer implies all integers?

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be rational numbers. Suppose that $a_1^k+a_2^k+\ldots+a_n^k$ is an integer for all positive integers $k$. Is it true that $a_1,a_2,\ldots,a_n$ ...
0
votes
0answers
19 views

Hilbert series computation for Hilbert scheme of $n$ points on $\mathbb C^2$

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{character}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $T$ acts on $x,y$ as $(t_1,t_2)$? ...
10
votes
1answer
245 views

Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

I'm reading the introductory bits in Procesi's Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for ...
0
votes
3answers
78 views

System of equations (contest problem)

Compute the ordered triple $(x,y,z)$ of positive real numbers that satisfies all three of the equations: $xy+x+y=19$ $yz+y+z=29$ $xz+x+z=53$ Please show me specific work and explain the law or ...
4
votes
1answer
98 views

Solving systems of equations

I had a system of equations and i want know the perfect method to solve that: Solve for $X, Y, Z$ where : $\\$ $X^² = Y + a$ $Y^² = Z + a$ $Z^² = X + a$
3
votes
0answers
32 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
5
votes
8answers
360 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
2
votes
2answers
42 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 + ...
4
votes
2answers
456 views

Need help solving a particular system of non-linear equations analytically

How would one go about analytically solving a system of non-linear equations of the form: $a + b + c = 4$ $a^2 + b^2 + c^2 = 6$ $a^3 + b^3 + c^3 = 10$ Thanks!
0
votes
0answers
25 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 ...
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
1answer
73 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
0answers
33 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 ...
15
votes
12answers
1k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
0
votes
0answers
26 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
80 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
47 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 ...
2
votes
0answers
28 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
34 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
176 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
410 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 ...
6
votes
0answers
109 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*} ...
1
vote
0answers
42 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
506 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
53 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 ...
9
votes
1answer
168 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 ...
4
votes
0answers
70 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
29 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
98 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
102 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?
0
votes
1answer
42 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 ...