Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
9
votes
0answers
65 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
15 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
$$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
9
votes
8answers
261 views
How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations
I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
0
votes
1answer
27 views
How to prove this symmetric polynomial equations?
I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}%
\frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1.
I tried to multiply the left of equation with $\Pi _{1\leq ...
1
vote
0answers
23 views
Are the coefficients of $\text{minpoly}(\alpha + \beta)$ polynomials in the coefficients of $\text{minpoly}(\alpha)$ and $\text{minpoly}(\beta)$?
Suppose you are given an algebraic field extension $L \supset K$ and $\alpha,\beta \in L$ with $f(X) = \text{minpoly}_K(\alpha)(X)=a_0+...+a_{m-1} X^{m-1}+X^m$ and ...
6
votes
2answers
104 views
A simple 2 grade equations system
If we have:
$$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
58 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
127 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
53 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
35 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
32 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, ...
0
votes
0answers
41 views
System of equations and Abel theorem
Consider this system of 3 equations to be solved in x,y and z:
$a x^m=(y+z)^n$
$by^m=(x+z)^n$
$cz^m=(x+y)^n$
The parameters $(a,b,c)$ and the unknown $(x,y,z)$ are all in $ℝ₊$. Also, m and n are ...
2
votes
0answers
60 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
32 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
2
votes
0answers
77 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 ...
2
votes
0answers
69 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
64 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
106 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
92 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
39 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
81 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
58 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
43 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
90 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
45 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
45 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 ...
6
votes
0answers
139 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
117 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
136 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
67 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
56 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
59 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
60 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 ...
4
votes
0answers
116 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
435 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
53 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
87 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
40 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
220 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
95 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
103 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 ...
3
votes
0answers
44 views
Existence of a Root of Elementary Monomials
Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$.
For example:
$$
m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3
$$
...
0
votes
0answers
42 views
Value of Elementary Symmetric Polynomials on a Geometric Progression
Is there a nice expression for the coefficients of $\prod_{i=0}^{n} (x-a^i)$, which are (up to sign) the elementary symmetric polynomials evaluated on a geometric progression?
2
votes
1answer
119 views
Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$
Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$,
$$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$
That is, the ...
0
votes
0answers
64 views
Showing that an alternating polynomial is the product of some symmetric polynomial and the Vandermonde polynomial
For simplicity, consider polynomials of two variables. Let $f(x,y)$ be an arbitrary alternating polynomial. I want to show that $f(x, y)$ is the product of some symmetric polynomial and the ...
7
votes
2answers
175 views
Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?
User Eric Gregor and I were talking in chat and he mentioned this question and postulated the possibility of an approach through symmetric polynomials. After some thinking, I came to this:
...
5
votes
0answers
119 views
Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?
Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
3
votes
0answers
128 views
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 ...
3
votes
1answer
249 views
Character of the $n^{\text{th}}$ symmetric power of the standard representation of $S_3$
So I am working out Fulton-Harris's Representation Theory text. For $S_3$, there is the standard representation $V$ which is two dimensional. That's all great and fine. Let $Sym^k(V)$ be the symmetric ...
6
votes
1answer
382 views
Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
$\alpha _n ^n-1=0$
$\alpha _n=e^{2 \pi i/n}$
$$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$
I have read in Jim Brown's paper on page 5 that ...
