Tagged Questions

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

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Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
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Explain the terms : homogeneous , symmetric , anti-symmetric , cyclic with respect to polynomials.

In answers to some of my previous questions , a lot of people used the terms homogeneous polynomial ( in a,b,c ) (under permutations of variables ) , cyclic polynomial ( in a,b,c) (under permutations ...
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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 ...
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Are the coefficients of $\text{minpoly}(\alpha + \beta)$ polynomials in the coefficients of $\text{minpoly}(\alpha)$ and $\text{minpoly}(\beta)$?

Suppose you are given an algebraic field extension $L \supset K$ and $\alpha,\beta \in L$ with $f(X) = \text{minpoly}_K(\alpha)(X)=a_0+...+a_{m-1} X^{m-1}+X^m$ and ...
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A simple 2 grade equations system

If we have: $$x^2 + xy + y^2 = 25$$ $$x^2 + xz + z^2 = 49$$ $$y^2 + yz + z^2 = 64$$ How do we calculate $$x + y + z$$
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Decomposition of products of monomial symmetric polynomials into sums of them

I'm trying to make sense of the answer given in: this question I am stuck at the phrase 'where the partitions $\gamma$ result from adding, respectively, from $\alpha$ all distinct partitions obtained ...
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Derivative of Schur function

In his answer to http://mathoverflow.net/questions/129854, R. Stanley says that the partial derivative (over the relevant x[i]) of the Schur function of a partition lambda of n equals the sum the ...
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Galois Theory References

This may not initially be a well posed question, but I'm looking for a good reference on Galois theory that covers it from the viewpoint of the symmetry in roots of an irreducible polynomial and not a ...
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$p(x)$ be a polynomial over $\mathbb{Z}$. If $P(a)=P(b)=P(c)=-1$ with integers $a,b,c$.Then $P(x)$ has no integral roots

Let $\mathbb{P}(x)$ be a polynomial over $\mathbb{Z}$. If $\mathbb{P}(a)$=$\mathbb{P}(b)$=$\mathbb{P}(c)$=$-1$ with integers $a,b,c.$ Then $\mathbb{P}(x)$ has no integral roots
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Understanding the Fundamental Theorem of Symmetric Polynomials within the context of proving $\pi$ transcendental

I am currently studying the proof of the transcendence of $\pi$. There are a bunch of proofs scattered across the web (here, here, and here, to list some); some derive from the Lindemann-Weierstrass ...
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Proofs of The Fundamental Theorem of Symmetric Polynomials

I have been considering a few proofs of this theorem, and I noticed that a few of them (for example Proof 1, and Proof 2) prove the theorem first for homogeneous symmetric polynomials and then ...
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Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?

I am trying to find a translation of this paper either in English or French (preferably English). I am not very optimistic, but i thought of asking in case somebody is more resourceful :)
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I'm not sure if I understand what the following question is asking: Show that the solution of the general biquadratic equation $x^{4}+ax^{3}+bx^{2}+cx+d=0$ can be obtained directly, that is, ...
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Equal-variables problem in three variables

This question resembles Vasile Cirtoaje’s equal-variables results, as explained here (although those results may be useless in the present problem). Let $s,p$ be two positive numbers with ...
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Choose the variables so that the weighted symmetric polynomial is minimal.

I've been struggling with the following problem for hours: Consider the expression $p^2\frac{x}{y+z}+q^2\frac{y}{x+z}+r^2\frac{z}{x+y}$, where $p,q,r>0$ are parameters. Choose $x,y,z\ge0$ so that ...
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'Galois Resolvent' and elementary symmetric polynomials in a paper by Noether

In Emmy Noether's 1915 paper "Der Endlichkeitssatz der Invarianten endlicher Gruppen", I saw the notion of a 'Galois resolvent', which I don't quite understand. Google didn't really help me with that, ...
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Reference request on symmetric polynomials

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$). I know very little about these polynomials. I've just noticed this odd ...
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Symmetrization of Powersum polynomials

Let $n\in\mathbb{N}$. Then for $i\in\mathbb{N}$ the $i-$th power sum if defined to be $p_i^{(n)}:=\sum_{j=1}^n x_j^i$. Then let $\lambda:=(\lambda_1,\ldots,\lambda_l)$ be a partition of $d$. We can ...
Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...