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

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0
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2answers
65 views

System of equations symmetric

How do I solve the following system of equation? $$ xyz = x+y+z $$ $$ xyt = x+y+t $$ $$ xzt = x+z+t $$ $$ yzt=y+z+t $$ I have no idea how to do.
3
votes
1answer
58 views

Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ whenever $p$ is prime?

Let $S_i(x_1,x_2,\dots,x_n)$ denote the $i$th elementary symmetric polynomial in $n$ variables. Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ from $0$ to $(p-2)$ whenever $p$ is ...
0
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1answer
53 views

Product as the sum of powers times a symmetric polynomial: What's the name of this property and what is it used for?

I noticed that the product of a group of positive integers $N$ with $n$ elements can be expressed as the sum of powers of the smallest member of $N$ times some (what I later found out be called) ...
6
votes
0answers
74 views

Integer divisibility

Given a (not strictly) decreasing sequence of natural positive numbers $a_1, a_2, \dots, a_n$ prove that $$ \prod_{i<j} j-i \quad\big|\quad \prod_{i<j} a_i - a_j - i +j $$ I already know a ...
2
votes
0answers
21 views

Symmetrized monomials under Weyl group?

Consider a given partition $\lambda=(\lambda_1,\lambda_2,...,\lambda_N)$ and start with the monomial $$z_1^{\lambda_1}z_2^{\lambda_2}...z_N^{\lambda_N}$$ in $N$ variables $z_1,z_2,...,z_N$. Now we ...
5
votes
1answer
104 views

Polynomials with $S_n \times \mathbb{Z}_2$ symmetry

Suppose that a polynomial $p(x_1\ldots x_n, y_1\ldots y_n)$ in $2n$ variables is invariant under the following operations: 1) $p(x_1\ldots x_n, y_1\ldots y_n)=p(y_1\ldots y_n, x_1\ldots x_n)$ 2) $\...
1
vote
3answers
86 views

How to solve this equation algebraically [closed]

Solve the following simultaneous equations on the set of real numbers: \begin{cases}x^2 + y^3 = x+1 \\ x^3+y^2=y+1\end{cases} Thanks for helping!
2
votes
2answers
46 views

What are the one-dimensional elements in the ring of symmetric functions?

The verification principle for $\lambda$-rings says (if I'm understanding correctly) that if you have a $\lambda$-ring $A$, and an equation using only $\lambda$-ring operations (addition, ...
1
vote
1answer
61 views

Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
1
vote
2answers
32 views

Solving system of non-linear equations.

So I'm trying to find the stationary points for $$f(x,y,z) = 4x^2 + y^2 +2z^2 -8xyz$$ Setting the partial derivatives to zero leads to: $$x-yz=0 \\ y-4xz=0\\z-2xy=0$$ Substiting $z=2xy$ into the ...
5
votes
1answer
76 views

Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
2
votes
1answer
57 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
1
vote
1answer
38 views

Systems of equation

Find non-negative solutions of systems of equations: $$\begin{cases} x^2y^2+1=x^2+xy \\ y^2z^2+1=y^2+yz \\ z^2x^2+1=z^2+zx \end{cases} $$ My work so far: 1) $(1;1;1) - $ solution. 2) $(y^2-1)x^2-...
1
vote
1answer
103 views

Solve this systems to condition $3x^3(x+1)^2=2y^2(z+3)^3$

if $x,y,z$be postive real numbers, solve systems of this following equation $$ 3x^3(x+1)^2=2y^2(z+3)^3\tag{1}$$ $$3y^3(y+2)^2=2z^2(x+1)^3\tag{2}$$ $$3z^3(z+3)^2=2x^2(y+2)^3\tag{3}$$ My approach is ...
0
votes
1answer
17 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
1
vote
0answers
25 views

System of Nonlinear Equations (sum of powers)

I want to show the only solution to the following system of equations is the trivial one ($x_{i} = 0$). I don't know if this is true, but I think it should be. Let $x_{i} \in \mathbb{C}$ for $1 \le i ...
0
votes
0answers
17 views

Schur polynomials with variables raised to a fixed power

For a partition $\lambda$ let $s_{\lambda}(x_1,\dots,x_k)$ denote the Schur polynomial in $k$ variables associated to $\lambda$ (let's assume that $k$ is sufficiently large compared to $\lambda$ that ...
1
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1answer
42 views

A binomial symmetric sum

Denote \begin{align*} \text{Sym}_k(\textbf{x})=\sum_{i_1<\cdots<i_k}x_{i_1}\cdots x_{i_k} \end{align*} as the $k$th elementary symmetric sum in monomials $\textbf{x} = (x_1, \cdots, x_n)$. If ...
0
votes
0answers
33 views

Help with deriving Newton's Identities

I am trying to derive Newton's identities for symmetric sums, namely in the case where $k > n \geq 1$ \begin{equation} \sum_{i=k - n}^k (-1)^{i + 1}S_{k-i}P_i = 0 \end{equation} where $S_k$ is the $...
1
vote
1answer
51 views

What are my polynomials called? (and where can I read about them)?

I've run across some polynomials that are natural enough I imagine must have been named and studied, but don't know what they're called. In the ring $\mathbb{Z}[x_i,y_i : i \in I]$ one has ...
2
votes
1answer
47 views

nth power symmetric polynomial in terms of Schurs polynomial

The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
0
votes
1answer
16 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
1
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0answers
34 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
3
votes
2answers
53 views

The Fundamental Theorem of Symmetric Polynomials

This theorem stays that any symmetric polynomial can be expressed as a polynomial of elementary polynomials. So let's suppose I have a polynomial $f(x_1,x_2,...,x_n)$ in $R[\mathbb{X}]$. I can find a ...
1
vote
1answer
69 views

Solve the system of equations $\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$

Find the number of real solutions to the system of equations $$\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$$ Let $f(x) = x^3-3x$ then for $x\in \mathbb{R}-(-2,2)$ we have $x_1 < ...
3
votes
1answer
75 views

Is it possible to “depress” any term in a polynomial with a suitable substitution?

If we have a degree $n$ polynomial $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$$with coefficients in a field, say $\Bbb C$, for concreteness, it is well known that the substitution $y= x + \...
3
votes
3answers
62 views

Solving system of three quadratic equations

$$\begin{cases} x^2 = yz + 1 \\ y^2 = xz + 2 \\ z^2 = xy + 4 \end{cases} $$ How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got ...
0
votes
2answers
56 views

Compute the trace of $\text{Sym}^2 \left(f \right)$ and that of $\text{Sym}^3 \left(f \right)$.

Consider the linear map $f: \mathbb{C}^3 \to \mathbb{C}^3$ defined by the matrix $$\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}.$$ Compute the trace of $\...
1
vote
4answers
116 views

Factorize $(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$

I am unable to factorize this over $\mathbb{Z}:$ $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$$ Since, this from an exercise of a book (E. J. Barbeau, polynomials) it must have a neat ...
2
votes
1answer
48 views

if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
0
votes
1answer
42 views

Efficient way to compute the symmetric reduction of special polynomials (specially for resolvents)

By the Fundamental Theory of Symmetric Polynomials every symmetric polynomial in $K[x_1, \dots, x_n]$ can be written uniquely in the elementary symmetric functions $s_1, \dots, s_n$. I know there are ...
0
votes
0answers
16 views

Is there any Newton type identity for the following ordered matrix symmetric polynomial?

I have the following sum of matrices $A_i,\ i=1,2,\cdots,\ n$ $$\sum_{n\ge i_1>i_2>\cdots>i_k\ge 1}A_{i_1}A_{i_2}\cdots A_{i_k}$$ This looks like an elementary symmetric polynomial but it is ...
0
votes
0answers
48 views

Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by $$\bigg(\sum_{k=0}^nx^k\bigg)^p~=~\frac1{(p-1)!}~\sum_{k=0}^{np}\frac{(n-|n-k|+p-1)!}{(n-|n-k|)...
2
votes
1answer
28 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le \...
2
votes
4answers
57 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
0
votes
0answers
17 views

Concave property on elementary symmetric polynomials

Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}} $ ...
0
votes
2answers
67 views

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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0answers
61 views

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 ...
0
votes
1answer
47 views

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 ...
0
votes
1answer
44 views

Number of solutions for system of elementary symmetric functions?

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ^{n}-e_{1}(X_{1},\ldots ,...
3
votes
4answers
87 views

Solving Symmetrical Equations Algebraically

I'm doing some Cambridge STEP papers and have come across a tricky set of equations. \begin{align*} 99 &= c^3 + 6 cd^2 \tag{1} \\ 70 &= 3c^2d + 2d^3 \tag{2} \end{align*} From looking ...
9
votes
1answer
154 views

Geometry of Elementary Symmetric Polynomials

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ^{n}-e_{1}(X_{1},\ldots ,...
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0answers
55 views

An Elementary Solution to a Polynomial Problem?

The following problem is from Larson's problem solving through problems: If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ \frac{b^{...
0
votes
1answer
142 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
1
vote
2answers
58 views

How to solve system: x_1+x_2+…+x_n=a [closed]

How to solve this system: $$\left\{\begin{matrix} x_1 & + &x_2 & + & \ldots & + & x_n &= & a \\ x^2_1& + &x^2_2 &+ & \ldots & + &x^2_n &=...
1
vote
4answers
54 views

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 ...
4
votes
0answers
30 views

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 ...
3
votes
1answer
72 views

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 ...
2
votes
1answer
87 views

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 ...
2
votes
1answer
74 views

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 ...