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

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Homework: Sum of the cubed roots of polynomial

Given $7X^4-14X^3-7X+2 = f\in R[X]$, find the sum of the cubed roots. Let $x_1, x_2, x_3, x_4\in R$ be the roots. Then the polynomial $X^4-2X^3-X+ 2/7$ would have the same roots. If we write the ...
4
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0answers
31 views

Arnold's combinatorial description of entropy.

V.I. Arnold says that entropy is related to the asymptotic behaviour of polynomial coefficients. This is mentioned in his book "Dynamics, Statistics and Projective Geometry of Galois Fields". Here ...
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1answer
73 views

Symmetry planes in spherical harmonic basis

Suppose I have a function $f(x):S^2\rightarrow\mathbb{C}$ in the degree four spherical harmonic basis: $$f(\theta,\varphi):=\sum_{k=-4}^4a_kY_4^k(\theta,\varphi).$$ I have two related questions: Is ...
2
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1answer
88 views

How to solve this set of symmetric polynomial expressions

So there's this set of polynomial expressions with degree n=3: $$ \left\{ \begin{array}{c} x_1 + x_2 + x_3 = a \\ x_1^2 + x_2^2 + x_3^2 = b \\ x_1^3 + x_2^3 + x_3^3 = c \end{array} \right. $$ How to ...
2
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1answer
74 views

Representation of eigenvector product using matrix elements

Let $A$ be a $n \times n$ real matrix, $(\lambda_i, v_i)$ be the $i$-th (eigenvalue, eigenvector) of $A^T$, and $x(t)$ be a vector of $n$ functions $x_i(t)$. For $\frac{d x(t)}{dt}=A x(t)$, the ...
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1answer
52 views

System of three equations with lots of symmetry and 6 unexpected (?) solutions.

I'm interested in the system of equations: $a(b^2+c)=c(c+ab)$ $b(c^2+a)=a(a+bc)$ $c(a^2+b)=b(b+ac)$ It is easy to see that $a=b=c=t$ are solutions for all $t$, in fact these are the only real ...
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2answers
25 views

Possible values for this specific line of variables.

I have this line of numbers: xy + z = xz + y = yz + x I need to find out all the possible values of x, y and z in this equation. Thank you!:) My usual problem ...
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1answer
47 views

$(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials?

Is it possible to express $(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials ? What I tried was ...
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1answer
29 views

polynomials in terms of elementary symmetric polynomials

Let a polynomial of $2n$-variables be $$ f(x_1,\cdots,x_n,y_1,\cdots,y_n)=\prod_{i,j=1}^n(1+x_i+y_j). $$ Let the elementary symmetric polynomials be $\alpha_1=\sum_{i=1}^n x_i$, $\alpha_2=\sum_{i<j}...
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0answers
34 views

terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $ e_1,\cdots,e_n$ and $$ \mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n]. $$ We define a ...
3
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1answer
47 views

Where is this converging to on $y=x$?

Well I was playing with graphs and I started plotting equations as the following: $$\underbrace{x+y}_{degree=1}=1 \tag{1}$$ $$\underbrace{x^2+y^2+xy}_{degree=2}+\underbrace{x+y}_{degree=1}=1 \...
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4answers
113 views

What is the sum of the cube of the roots of $ x^3 + x^2 - 2x + 1=0$?

I know there are roots, because if we assume the equation as a function and give -3 and 1 as $x$: $$ (-3)^3 + (-3)^2 - 2(-3) + 1 <0 $$ $$ 1^3 + 1^2 - 2(1) + 1 > 0 $$ It must have a root ...
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4answers
115 views

How to solve this equation $\ x^2-y^2=29$ [closed]

How to find the solutions $\ (x,y)$ in this equations 1.$\ x^2-y^2=29$ 2.$\ x^2+xy+y^2=0$ any tricks plz
2
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1answer
62 views

Solution to the following set of equations

Is the solution to the following: $$a^2+b^2=1$$ $$c^2+d^2=1$$ $$ad+bc=1$$ still $a=d=\cos z$, $c=-b=\sin z$, when $a,b,c,d \in \mathbb C$?
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2answers
60 views

Proof for $A,B \in M_n(\mathbb{F})$ that if $[A,B]=tA$ for $0\neq t\in\mathbb{F}$, then $A^n=0$ [duplicate]

Statement. Suppose we have a square matrices $A,B$ of order $n$ over a field $\mathbb{F}$ of characteristics $0$ or $p>n$. If $[A,B]=AB-BA=tA$ for some nonzero $t\in\mathbb{F}$, then $A^n=0$. The ...
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1answer
79 views

Proof that if $\mathrm{tr}\,A^k=0$ for all $k=1,\ldots, n$, then $A^n = 0$ [duplicate]

Statement. Suppose we have a square matrix $A$ of order $n$ over a field $\mathbb{F}$ of characteristics $0$ or $p>n$. There is a theorem that if $\mathrm{tr}\,A^k=0$ for all $k=1,\ldots, n$, then $...
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1answer
76 views

System of polynomial equations with cyclic symmetry

solve a system of equations with unknowns x1, x2, xn where n >=2
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2answers
63 views

Number of monomial symmetric polynomials in three variables

I am trying to find a reference for a formula regarding the number of monomial symmetric polynomials of degree $m$, in three variables. I believe that this number is given by $1+\left \lfloor{\frac{m^...
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0answers
77 views

Symmetrical Non-linear Constrained Equation

How would you approach this problem? I need all the possible solutions $$ (x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4) $$ which satisfy $$ \frac{(x_1-x_2)^2-(y_1-y_2)^2}{(x_1-x_2)^3(y_1-y_2)^3}+\frac{(x_1-x_3)^...
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5answers
143 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:)
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1answer
116 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
42 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
69 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 $$(x^2+x+1)(y^2+y+1)(...
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1answer
52 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
39 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 $...
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1answer
51 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
94 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 \sqrt{\...
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0answers
95 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 ...
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1answer
43 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 $P=...
3
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1answer
42 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 ...
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1answer
77 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 ...
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2answers
53 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 ...
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1answer
70 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 contradiction: ...
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0answers
48 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
33 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
48 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 ...
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1answer
47 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
118 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: ...
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1answer
28 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 $Q\in R[X_1,...,X_n]$ such ...
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0answers
35 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
32 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 ...
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0answers
26 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
52 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 $$(x-...
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0answers
46 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 ...
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1answer
147 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
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1answer
83 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 i_1<i_2\cdots<i_k\leq4}\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{...
2
votes
2answers
165 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 + x(...
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 $s_3$.
2
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1answer
69 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 $y=(y_1,y_2,\...
1
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0answers
65 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 + z_{n}...