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

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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
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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
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1answer
78 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 ...
2
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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 ${x_n=F(\{S_p\})}$? ...
4
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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 ...
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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: $$ a \cdot b = 3 \cdot a - b + 1 $$ $$ b \cdot c = 3 \cdot b - c + 1 $$ $$ c \cdot a = 3 \cdot c - a + 1 $$
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1answer
32 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
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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 ...
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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 ...
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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
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1answer
35 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 ...
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 ...
1
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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 ...
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
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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 ...
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 $n$ ...
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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 ...
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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 ...
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 ...
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1answer
55 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: ...
2
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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 $Q\in R[X_1,...,X_n]$ ...
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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 ...
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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 ...
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1answer
27 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
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0answers
19 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 ...
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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
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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 ...
5
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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
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2answers
88 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 + ...
4
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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 $s_3$.
2
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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 ...
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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 + ...
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4answers
254 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 ...
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0answers
29 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 ...
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1answer
41 views

Splitting a polynomial into parts which are symmetric and antisymmetric under exchange of variables.

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 three variables (and some further ...
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1answer
32 views

Splitting a polynomial into a symmetric and an antisymmetric part.

How can I split a polynomial into a symmetric and an antisymmetric part? I have an explicit polynomial, which is a function of three variables (and some further constants). The symmetry properties ...
0
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0answers
50 views

fundamental theorem of symmetric polynomials

Let $P_n(a,b,c)$ be a polynomial of variables $a,b,c$. By Newton's fundamental theorem of symmetric polynomials, there is a unique $P_n$ such that $$ x^n+y^n+z^n=P_n(x+y+z, x^2+y^2+z^2,x^3+y^3+z^3). ...
9
votes
1answer
144 views

On an identity with $n$ variables

We know the followings : $$\color{red}{{x_1}^2+{x_2}^2-2x_1x_2}=\color{blue}{(x_1-x_2)^2}$$ ...
1
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1answer
55 views

What is a symmetric polynomial?

I'm reading about symmetric polynomials at the moment, and came upon this statement: Let $G(x_1, \ldots, x_n)$ be a symmetric polynomial. Separate out $x_n$, representing $G$ as follows: $G_\nu ...
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1answer
74 views

How to solve the system of 4 equations of four unknowns

Solve this system of the four equations of four unknowns $a, b, c, d>0 $ $$ 165(a+b+c)=abc\tag1 $$ $$220(a+b+d)=abd \tag2 $$ $$297(a+c+d)=acd\tag3 $$ $$540(b+c+d)=bcd \tag4 $$ I tried to ...
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1answer
36 views

Newton polynomials

Consider the family of symmetric polynomials $\sum^n_{i=1} x_i^k\in\mathbf{Z}[x_1,\ldots,x_n]$. By the fundamental theorem on symmetric polynomials there is a unique Newton poylnomial ...
2
votes
0answers
35 views

Find the coefficients in $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$

Use evaluation homomorphisms $F[x_1,x_2, \dots, x_n] \to F$ to obtain the coefficients in: $$(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$$ where the $s_i$ ...
2
votes
3answers
68 views

Solving $n$ unknowns with $n$ independent equations

Is it always possible to solve $n$ independent equations with $n$ unknowns? Or is it possible to solve the following 3 equations with 3 unknowns? $$x + y + z = a$$ $$x^2 + y^2 + z^2 = b$$ $$x^3 + ...
2
votes
1answer
48 views

Why is the inverse of this matrix expressible in terms of elementary symmetric polynomials?

Suppose we have $N$ distinct real numbers $x_1, x_2, \ldots, x_N$ and consider the $N \times N$ matrix $$P = \left[\! \begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1 \\ x_1 & x_2 & ...
0
votes
0answers
20 views

counting how many entries from a given Young tableau contribute to hook length made from two differtn YTs

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ and define for $s=(i,j)\in R$ (it makes also sense for $s$ outside ...
1
vote
1answer
23 views

A question on a proof concerning resultants I don't quite get

http://planetmath.org/proofthatsylvestersmatrixequalstheresultant Heres a link to a proof I found concerning the relation between Sylvesters matrix and resultants. Most of it makes sense. I do have ...
3
votes
2answers
71 views

System of equations with 2 parameters

I have no idea how even to start! \begin{align*} (u^2+v^2)(u+v)&=15uv \\ (u^4+v^4)(u^2+v^2)&=85u^2v^2 \end{align*}
3
votes
1answer
61 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...