# Tagged Questions

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

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### symmetric polynomials and the Newton identities

I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
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### Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
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### Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

I'm reading the introductory bits in Procesi's Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for ...
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### Schur skew functions

Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ ...
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### Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
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### 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 ${x_n=F(\{S_p\})}$? ...
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### 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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### 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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### A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$\frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1)$$ My first question is, how can I prove this? It seems trivial ...
### Factoring the group action on $\prod X_k^k$ gives another group?
Let's say you have a monomial symmetric polynomial, like the following $$m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 +X_2^1X_1^2X_3^3X_4^4 + \text{all permutations...}$$ Then you can ...