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

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11
votes
1answer
159 views

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 ...
0
votes
1answer
298 views

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 ...
0
votes
1answer
53 views

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 ...
8
votes
0answers
194 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots ...
1
vote
0answers
42 views

Clarification on proof including symmetric polynomials

This is regarding theorem 3 in this article. My problems begin after the equalities denoted by (5). My problems aren't so much about theory really, I think. I'm disregarding the authors' notation a ...
0
votes
0answers
73 views

Is there a subgroup of $S_{10}$ having $5040$ elements other than $S_7$?

I'm trying to generate to monomial symmetric polynomial $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 + \text{all permutations,} $$ starting from the first element by applying all ...
2
votes
0answers
79 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
1
vote
0answers
44 views

Some lemma on elementary symmetric polynomial

$$e_t(x_2,\ldots,x_n) = \sum^{n-t}_{i=1} (-1)^{i+1} \frac{e_{t+i}(x_1,\ldots,x_n)}{x^i_1}\text{ for every } 0\leq t < n.$$ $e_t$ is the $t^{th}$ elementary symmetric polynomial in the variabel ...
1
vote
1answer
71 views

symmetric polynomial inequality?

I put $n\ge 2$ balls of various sizes into an urn. I draw two balls (without replacement) from the urn. With each draw, I draw any given ball with probability proportional to its size. Can you ...
2
votes
0answers
52 views

Expansions of symmetric polynomials in terms of Jack symmetric polynomials

I was wondering if someone could help me with some Jack polynomial calculations. (I use the notation of I.G. Macdonald's book "Symmetric Functions and Hall Polynomials") Those of you familiar with ...
34
votes
2answers
533 views

What is the function space generated by addition and $(a,b)\mapsto (a+b)^{-1}\cdot a\cdot b$ of elements and their inverses?

(the motivation section turned out a little long, the mathematical question is at the end) I need to work with electrical circuts at the moment, computing effective impedances etc. From ...
1
vote
0answers
26 views

signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
14
votes
11answers
1k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
0
votes
1answer
51 views

How to prove this symmetric polynomial equations?

I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}% \frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1. I tried to multiply the left of equation with $\Pi _{1\leq ...
1
vote
0answers
40 views

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 ...
7
votes
2answers
142 views

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 $$x + y + z$$
1
vote
1answer
165 views

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 ...
5
votes
2answers
185 views

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 ...
0
votes
2answers
241 views

Solution to a system of symmetric equations

After applying the Lagrange multiplier method, I got the following system of equations, which is quite symmetric: $(x+y)^2 + (x+z)^2 = \frac{2}{3} \lambda x$ $(y+x)^2 + (y+z)^2 = \frac{2}{3} \lambda ...
3
votes
1answer
45 views

How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$

I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
1
vote
2answers
94 views

Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$

Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
2
votes
0answers
134 views

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 ...
1
vote
1answer
42 views

$p(x)$ 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
4
votes
2answers
237 views

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 ...
4
votes
0answers
395 views

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 ...
4
votes
0answers
74 views

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 :)
2
votes
0answers
288 views

general biquadratic equation

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, ...
1
vote
1answer
119 views

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 ...
0
votes
1answer
44 views

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 ...
6
votes
0answers
174 views

'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, ...
2
votes
0answers
81 views

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 ...
1
vote
0answers
75 views

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 ...
3
votes
3answers
136 views

Ring of invariants of Klein Four group

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 ...
2
votes
1answer
63 views

What is the degree of the fourier expansion

Let $ f:\{-1,1\}^3 \rightarrow \{-1,1\} $ , $f(x)= \operatorname{sgn}(x_1+x_2+x_3)$; (Majority function), then Fourier expansion of $f$ is $f(x)= \frac{1}{2} ...
1
vote
1answer
56 views

Counting primes of the form $S_1(a_n)$ vs primes of the form $S_2(b_n)$

Let $n$ be an integer $>1$. Let $S_1(a_n)$ be a symmetric irreducible integer polynomial in the variables $a_1,a_2,...a_n$. Let $S_2(b_n)$ be a symmetric irreducible integer polynomial in the ...
7
votes
0answers
163 views

Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials : $$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq ...
1
vote
2answers
166 views

Symmetric inequality on positive numbers whose product is one

Despite many attempts, no one at StackOverflow has succeeded in solving that old question about proving a deceptively simple-looking inequality. I propose now a weaker and slightly simpler inequality ...
3
votes
2answers
236 views

Roots of power sum symmetric polynomials

I had a few questions about the roots of power sum symmetric polynomials: Given that $x_1^k+x_2^k+x_3^k= 0$ for all $k \not \equiv 0\mod 3$ and is non-zero otherwise, if we assume none of the ...
2
votes
1answer
135 views

Symmetric polynomial optimization

Recently I asked a stupid question here (there’s no harm in that, even Fields medalist Terence Tao advises to ask dumb questions once in a while). Here is a variant question that may be more ...
1
vote
1answer
68 views

Path connectedness of a particular algebraic set

Let $n\geq 3$, let $a$ and $b$ be two positive numbers, and $$ \Omega = \bigg\lbrace (x_1,x_2, \ldots ,x_n) \in ]0,+\infty[^n \ \bigg| \ x_1+x_2+ \ldots +x_n=a, \ x_1x_2 \ldots x_n=b \bigg\rbrace $$ ...
3
votes
1answer
90 views

Optimize a symmetric polynomial on a compact set

This looks like a stupid question, but the obvious answer (if there is one) eludes me … Let $f(x,y)$ be a symmetric polynomial in $x$ and $y$. Then $f$ attains a minimum $m$ on the compact set ...
2
votes
4answers
102 views

expressing some polynomial in terms of symmetric polynomials

Express the element $(a-b)^2(a-c)^2(b-c)^2$ In terms of the symmetric elementary polynomials. I read the proof using Galois Theory, that any symetric polynomials can be written in terms of the ...
5
votes
0answers
135 views

Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?

Let me start with the following on elementary symmetric polynomials: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ...
16
votes
3answers
474 views

A generalized (MacLaurin's) average for functions

The average value of a function $y=f(x)$, on an interval $[a,b]$, is ${1\over {b-a}}\int_a^b f(t)dt$. This of course relates to the arithmetic average. It is easy to see that a corresponding formula ...
2
votes
0answers
66 views

Free software for expresing a resolvent as function of coefficients

This relates to question "Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?" I would like to try absolute resolvent for group $C_5$ in $S_5$. For example ...
1
vote
2answers
157 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
3
votes
0answers
42 views

Symmetrizing a sequence of vectors

Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
4
votes
1answer
348 views

A proof of the fundamental theorem of symmetric polynomials

I'm reading Exploratory Galois Theory by John Swallow. On page 123 he gives the following remark / alternate proof of the fundamental theorem of symmetric polynomials: Let $K$ be a field and $L$ ...
4
votes
1answer
121 views

Analog of Newton's theorem for symmetric polynomials

Newton's theorem of symmetric polynomials says that every symmetric polynomial can be written as a polynomial in elementary symmetric polynomials. Hence when $S_n$ acts on $\mathbb{Q}(x_1,...,x_n)$ ...
2
votes
1answer
173 views

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 ...