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

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Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by ...
2
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1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
2
votes
4answers
47 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
0
votes
0answers
12 views

Concave property on elementary symmetric polynomials

Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}} $ ...
0
votes
2answers
61 views

Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$. What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different? Edit: How can I show that ...
1
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0answers
43 views

Center of the group algebra of the symmetric group

How to prove that the center of the group algebra of the symmetric group is generated by 1-cycle conjugacy classes? I mean, that the center (consisting on class functions) is multiplicatively ...
0
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1answer
43 views

prove a polynomial identity..

The equation is that $h_m(x_1, \cdots, x_n, a)-h_m(x_1, \cdots, x_n, b)=(a-b)h_{m-1}(x_1, \cdots, x_n, a, b)$ where $h_m$ is a complete homogeneous symmetric polynomial. See and find several ...
0
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1answer
39 views

Number of solutions for system of elementary symmetric functions?

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ...
3
votes
4answers
82 views

Solving Symmetrical Equations Algebraically

I'm doing some Cambridge STEP papers and have come across a tricky set of equations. \begin{align*} 99 &= c^3 + 6 cd^2 \tag{1} \\ 70 &= 3c^2d + 2d^3 \tag{2} \end{align*} From looking ...
5
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1answer
63 views

Geometry of Elementary Symmetric Polynomials

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ...
1
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0answers
53 views

An Elementary Solution to a Polynomial Problem?

The following problem is from Larson's problem solving through problems: If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ ...
0
votes
1answer
71 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
1
vote
2answers
56 views

How to solve system: x_1+x_2+…+x_n=a [closed]

How to solve this system: $$\left\{\begin{matrix} x_1 & + &x_2 & + & \ldots & + & x_n &= & a \\ x^2_1& + &x^2_2 &+ & \ldots & + &x^2_n ...
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4answers
50 views

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
27 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 ...
3
votes
1answer
56 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
votes
1answer
83 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
votes
1answer
58 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
51 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
24 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 ...
0
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1answer
44 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 ...
1
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1answer
25 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 $\alpha_1=\sum_{i=1}^n x_i$, ...
2
votes
0answers
21 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
votes
1answer
45 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
101 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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votes
4answers
95 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
votes
1answer
60 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 $a,b,c,d \in \mathbb C$?
2
votes
2answers
55 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 ...
1
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1answer
66 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
66 views

System of polynomial equations with cyclic symmetry

solve a system of equations with unknowns x1, x2, xn where n >=2
2
votes
2answers
57 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 ...
1
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0answers
68 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 $$ ...
2
votes
5answers
142 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
104 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
votes
1answer
35 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
votes
2answers
66 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
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1answer
51 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
37 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
49 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
93 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
73 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
41 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
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1answer
35 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
74 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
votes
1answer
69 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
45 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
26 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 ...
0
votes
1answer
46 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
38 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 ...