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

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11
votes
1answer
182 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
0answers
23 views

If $f-g$ is a symmetric polynomial, each term is divisible by $X_1,\dots,X_{n-1}$?

The following comes from Bourbaki's Algebra II, Chapter IV Section 6, Theorem 1. Here the $s_i$ are the elementary symmetric polynomials in $X_1,\dots,X_n$, and $E=A[X_1,\dots,X_n]$, and $A$ is an ...
2
votes
5answers
135 views

Nice algebra question

The real numbers $x,y,z$ satisfy $$x+y+z=4$$ $$xy+yz+zx=2$$ $$xyz=1$$ Then $x^{3}+y^{3}+z^{3}$=??. It's for sharing a new ideas, thanks:)
3
votes
1answer
79 views

Decomposition of polynomial ring as $S_n$-module

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...
3
votes
4answers
85 views

If $x,y,z \in \Bbb{R}$ such that $x+y+z=4$ and $x^2+y^2+z^2=6$, then show that $x,y,z \in [2/3,2]$.

If $x,y,z\in\mathbb{R}$ such that $$x+y+z=4,\quad x^2+y^2+z^2=6;$$then show that the each of $x,y,z$ lie in the closed interval $[2/3,2]$. I have been able to solve using $2(y^2+z^2)\geq(y+z)^2$. Is ...
2
votes
1answer
25 views

Inverting power sum of symmetric polynomial

Suppose I have a set of power sum symmetric polynomial as $$S_p =\sum_i^N x^p_i ~~;~~~~~~~~p=\{1,N\}$$ and I have N of them $\{S_1...S_N\}$ Question is given this, can we find ${x_n=F(\{S_p\})}$? ...
1
vote
1answer
44 views

solve system equation: $ a \cdot b = 3 \cdot a-b+1, b \cdot c = 3 \cdot b - c + 1, c \cdot a = 3 \cdot c - a + 1$

I want to solve this system of equations but i'm stuck. Here is it: $$ a \cdot b = 3 \cdot a - b + 1 $$ $$ b \cdot c = 3 \cdot b - c + 1 $$ $$ c \cdot a = 3 \cdot c - a + 1 $$
4
votes
2answers
63 views

this inequality $\prod_{cyc} (x^2+x+1)\ge 9\sum_{cyc} xy$

Let $x,y,z\in R$,and $x+y+z=3$ show that: $$(x^2+x+1)(y^2+y+1)(z^2+z+1)\ge 9(xy+yz+xz)$$ Things I have tried so far:$$9(xy+yz+xz)\le 3(x+y+z)^2=27$$ so it suffices to prove that ...
1
vote
1answer
79 views

solve system equation: $ 2a^2 - 1 = b, 2b^2 - 1 = c, 2c^2 - 1 = a $

I have this system equation: $$ 2a^2 - 1 = b $$ $$ 2b^2 - 1 = c $$ $$ 2c^2 - 1 = a $$ From system equation we see that $ a \neq 0 , b \neq 0, c \neq 0 $ , so : $ 2a^2 - 1 \neq 0 => a \neq ...
0
votes
1answer
41 views

system of equations 3 variables

I should find $A,B$ and $C$. I know answers but can't figure out how to solve it. Anyone? We are to find value of $x^4+y^4+z^4$ when $x, y$ and $z$ are real numbers which satisfy the following ...
1
vote
1answer
33 views

Good Reason for Partitions Indexing Symmetric Functions?

I'm mostly unfamiliar with the study of symmetric functions. However, it's my understanding that: We are interested in, as a basic object, the vector spaces $\Lambda_n$ of symmetric polynomials in ...
1
vote
1answer
35 views

How does this relate to Vieta's formula?

I was reading this PDF: http://diendantoanhoc.net/forum/index.php?app=core&module=attach&section=attach&attach_id=219 On Page 2, the author mentions Vieta's Formula. Now I am familiar ...
3
votes
1answer
68 views

If $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real, then so are a,b,c

Let $a,b,c$ be complex numbers with distinct magnitudes such that $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real. Prove that $a,b,c$ are real numbers as well. I tried to go for ...
1
vote
0answers
40 views

An identity with determinant and trace of a matrix

How to prove the following identity: $$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A)$$ where $\mathrm{Tr}_{\sigma}(A)$ is defined as following if $\sigma$ is ...
1
vote
1answer
36 views

Symmetric Polynomials and Automorphisms of Complex Polynomial Rings

I asked a version of this question earlier, but it was very imprecise and poorly formatted, so I decided to create a new question. Suppose we have an ordered set of $n(n-1)/2$ distinct polynomials ...
1
vote
1answer
62 views

Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
3
votes
1answer
31 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 ...
2
votes
2answers
51 views

How to retrieve the expression of $a^3+b^3+c^3$ in terms of symmetric polynomials?

I recently had to find without any resources the expression of $a^3+b^3+c^3$ in terms of $a+b+c$, $ab+ac+bc$ and $abc$. Although it's easy to see that $a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)$, I couldn't ...
3
votes
0answers
28 views

A “nice” orthogonal basis for translation invariant symmetric polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
1
vote
1answer
56 views

Solving Systems of Equations ( Binomial * Trinomial )

This is not a homework question; rather a review for a Mechanical Engineering Board Exam. I need to find an efficient way to solve equations of these types: ...
1
vote
0answers
17 views

Divided power question.

Let $E$ be a free module, we define the $r$-th divided power as the dual of the symmetric power $D_r(E):=(S_r(E^*))^*$. For every $u \in E$ we can define its $r$-th divided power $u^{(r)} \in D_r$ by ...
7
votes
2answers
319 views

System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$.

$$a+b+c=4$$$$a^2+b^2+c^2=8$$ I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated. It's obvious that ...
4
votes
1answer
27 views

Generating function of symmetric power representation

Let $\rho:G\rightarrow GL(V)$ be a complex representation. For each $n$, let $\chi_{\text{Sym}^n}$ be the character of the n-th symmetric power of $V$. Prove for each $g\in G$, $$\sum_{i=0}^\infty ...
11
votes
4answers
1k views

Algorithm(s) for computing an elementary symmetric polynomial

I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing ...
1
vote
1answer
44 views

A system of simultaneous equations

I'm currently stuck solving this set of equations. $$x(x+y+z)=4-yz$$ $$y(x+y+z)=9-zx$$ $$z(x+y+z)=25-xy$$ Here's what I've done so far: By subtracting the second equation from the first, I got ...
2
votes
1answer
17 views

Where is “homogenuity” used in this proof?

Theorem Let $R$ be a ring and $P\in R[X_1,...,X_n]$ be a symmetric polynomial and $p_1,...,p_n$ be the elementary symmetric polynomials. Then, there exists a polynomial $Q\in R[X_1,...,X_n]$ ...
0
votes
0answers
7 views

Is my proof showing that if products of powers of elementary symmetric polynomials are the same, then the power is same, correct?

Below proof is assuming this lemma, which can be proven easily: Let $\leq$ be a monomial ordering on $R[X_1,...,X_n]$. Let $f,g\in R[X_1,...,X_n]$ be nonzero polynomials such that $LC(f)$ is ...
0
votes
0answers
24 views

Does the fundamental theorem of symmetric polynomials hold in any ring?

Fundamental theorem of Symmetric polynomials: Let $R$ be a commutative ring and $e_0,...,e_n$ be the elementary symmetric polynomials of $R[X_1,...,X_n]$. Let $\Phi:R[X_1,...,X_n]\rightarrow ...
1
vote
1answer
28 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
0
votes
0answers
20 views

Is there a natural link between symmetric polynomials and symmetric algebra?

Let $R$ be a commutative ring and $R[X_1,...,X_n]^{S_n}$ be the ring of symmetric polynomials. I have learned some basic properties of this ring and the results are really similar to those by ...
5
votes
1answer
134 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
3
votes
1answer
248 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
4
votes
1answer
168 views

Symmetric polynomials have unique expressions as polynomials in symmetric elementary functions

Let $s_i$ be the symmetric elementary functions. For example, $s_1=x_1+\cdots+x_n$. Suppose a polynomial $p(z_1,\ldots,z_n)\in R[z_1,\ldots,z_n]$ satisfies $p(s_1,\ldots,s_n)=0$ in ...
3
votes
0answers
130 views

Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.

Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
0
votes
1answer
160 views

Roots of elementary monomials

Let $m_\lambda(X_1,X_2,...X_N)$ be a monomial symmetric function with partition $\lambda$. For example: $$ m_{(3,1,1)}(X_1,X_2,X_3) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $$ Is there a general ...
3
votes
1answer
264 views

Which Newton's Identities are being referred to here?

I do not understand this line of Wikipedia's page on the Basel problem: If we formally multiply out this product and collect all the $x^2$ terms (we are allowed to do so because of Newton's ...
5
votes
1answer
131 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
3
votes
1answer
76 views

Inequality of elementary symmetric polynomials

Let $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ with $\lambda_i>0$ for $i=1,2,3,4$. Let $$\sigma_k(\lambda)=\sum_{1\leq ...
2
votes
0answers
37 views

Looking for the name of particular collection of polynomials

I came across the following algebraic structure when working on a seemingly unrelated problem and am unable to find a name for it. Let $R$ be a commutative ring with identity. Given ...
2
votes
1answer
64 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
2
votes
2answers
91 views

Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$

Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$ Suppose $a, b, c$ are roots of $P(x)$. $$P(x) = k(x - a)(x - b)(x - c)$$ But then I get $(k = 1)$ $$P(x) = x^3 - 12x^2 + ...
7
votes
4answers
180 views

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove ...
4
votes
1answer
35 views

Symmetric polynomials and g non symmetric

If $g=x_1+2x_2+3x_3, s_1=x_1+x_2+x_2, s_2=x_1x_2+x_1x_3+x_2x_3$ and $s_3=x_1x_2x_3$ , write $x_1, x_2$ and $x_3$ in function of $g, s_1, s_2$ and $s_3$.
9
votes
1answer
272 views

'Galois Resolvent' and elementary symmetric polynomials in a paper by Noether

In Emmy Noether's 1915 paper "Der Endlichkeitssatz der Invarianten endlicher Gruppen", I saw the notion of a 'Galois resolvent', which I don't quite understand. Google didn't really help me with that, ...
5
votes
9answers
422 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
3
votes
1answer
68 views

(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define ...
1
vote
0answers
64 views

System of (non linear) equations

Let $n \geq 2$. Could it be proved that the following system, with $z_k\in \mathbb C$, $ \begin{cases} z_1^n + z_{n}z_1^{n-1} + z_{n-1}z_1^{n-2} + \cdots + z_2z_1+z_1 & = 0 \\ z_2^n + ...
6
votes
4answers
255 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
0
votes
1answer
42 views

Algorithm for writing a symmetric polynomial P(x,y) in terms of x+y and xy

Assume I am given a polynomial of two variables by the list of its coefficients, $P(X,Y)=\sum_{i,j}C_{ij}X^i Y^j$, and that this polynomial is symmetric, i.e. $C_{ij}=C_{ji}$. I am looking for an ...
0
votes
0answers
30 views

Is every polynomial of two variables separable into a symmetric and antisymmetric part?

How can I split a polynomial into parts which are symmetric and antisymmetric under exchange of the variables? I have an explicit polynomial, which is a function of two variables (and some further ...