# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### $(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ 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 ...
### 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 ...