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

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4
votes
1answer
35 views

When is a recurrence the sum of the powers of the roots of a polynomial?

Newton's formula allows one to calculate the sum $S_n(P)$ of the $n$th powers of the roots of a given monic polynomial $P$ without finding the roots explicitly. (This works even when the roots ...
0
votes
0answers
9 views

Equality of sum of fractions implies correspondence of terms

I am working in a theorem of Jhonson and Newman about cospectrality and got stucked un this claim. can you help me? $a_i$ and $b_i$ are non negative numbers, $z\in\mathbb{C}$ and $d_i \neq d_j$ for $...
0
votes
0answers
32 views

Plaid in generic position. Counting faces.

I write $\pi_n$ to denote a group of $n$ parallel lines. Consider a family of $(\pi_1,\pi_2,\ldots,\pi_s)$ parallel groups each with $(n_1,n_2,\ldots,n_s)$ parallel lines. Arrange the family of ...
1
vote
0answers
21 views

Factorising Cyclic expression .

What are ways for factorising cyclic expressions? Note: I am not saying about specific one. Just ways of factorising cyclic expressions.
2
votes
2answers
48 views

Solving equations system: $xy+yz=a^2,xz+xy=b^2,yz+zx=c^2$

Solve the following system of equations for $x,y,z$ as $a,b,c\in\Bbb{R}$ \begin{align*}xy+yz&=a^2\tag{1}\\xz+xy&=b^2\tag{2}\\yz+zx&=c^2\tag{3}\end{align*} My try: Assume that $x,y,z\...
0
votes
2answers
25 views

Diffeomorphism of elementary symmetric polynomials

Let $\sigma_1(x,y,z) = x + y + z$, $\sigma_2(x,y,z) = xy + xz + yz$ and $\sigma_3(x,y,z) = xyz$. When is the map $\Phi: \mathbb{R}^3 \to \mathbb{R}^3, \Phi\,(x,y,z) = \begin{pmatrix} \sigma_1(x,y,z) \...
2
votes
0answers
70 views

Name these symmetric polynomials over non-commutative variables…

Given a set of $N$ non-commutative variables $x_k$. Is there a special name for symmetric polynomials of homogenous degree $d$ of the form that all $x_k$s appear with exponent at most $1$ at a time? ...
3
votes
1answer
103 views

Fixed points of polynomial ring homomorphism

$S=\mathbb R[x+y+z, xy+yz+zx, xyz]$ is the ring of the symmetric polynomials in $\mathbb R[x,y,z]$. Let $\psi\colon S \to R[x,y,z]$ be a ring homomorphism such that \begin{align} x &\mapsto -x,\\...
2
votes
1answer
52 views

Why weight of a monomial $X_1^{v_1}\cdots X_n^{v_n}$ is defined as $v_1+2v_2+\cdots + nv_n$, but not $v_1+\cdots+v_n$?

I was looking the proof of fundamental theorem on symmetric polynomials from Lang's Algebra (see this, p.190-192). I didn't understood one thing from proof and one question came to mind from statement ...
1
vote
0answers
23 views

Special case of Pieri-Rule

is there an "elementary" (read: short combinatorial) proof for the rule $$ s_\lambda \cdot s_{(1)} = \sum_{\mu} s_{\mu} $$ where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a ...
2
votes
0answers
57 views

Simple proof of Newton identities

The functions $s_1=x_1+x_2+\cdots +x_n$, $s_2=\sum_{i<j} x_ix_j$, $\cdots$, $s_n=x_1x_2\cdots x_n$ are elementary symmetric functions in $x_1,x_2,\cdots,x_n$ (or more precisely, elementary ...
1
vote
0answers
26 views

Symmetric sum of powers

Let $\mathbb C [x_1, \dots, x_n]$ be a ring of polynomials with complex coefficients. We then define the symmetric polynomials \begin{align*} S_0 &= 1 \\ S_1 &= x_1 + \cdots + x_n \\ S_2 &...
4
votes
1answer
197 views

How to decide if a polynomial is symmetric?

First, is the following: $$f=\frac{3}{5}(x_1^5 + x_2^5 + x_3^5 + x_4^5)-\frac{7}{12}(x_1^2x_2^2 - x_1^2x_3^2-x_1^2x_4^2-x_2^2x_3^2-x_2^2x_4^2-x_3^2x_4^2)$$ a symmetric polynomial? And, if yes, how do ...
0
votes
2answers
41 views

Symmetric roots of polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $x^3 - x^2 + 2x - 3$ $\in \mathbb{C}[x]$. Calculate $\alpha_1^3 + \alpha_2^3 + \alpha_3^3$. What to do here exactly? I already ...
0
votes
2answers
66 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
63 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
votes
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
80 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
23 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 ...
4
votes
1answer
105 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
87 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
65 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
33 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
78 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
58 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
39 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
27 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
19 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
vote
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
52 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
51 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
vote
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
57 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
70 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
76 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
63 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
57 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
43 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
17 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
20 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}}}} $ ...