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

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2
votes
1answer
15 views

Where is “homogenuity” used in this proof?

Theorem Let $R$ be a ring and $P\in R[X_1,...,X_n]$ be a symmetric polynomial and $p_1,...,p_n$ be the elementary symmetric polynomials. Then, there exists a polynomial $Q\in R[X_1,...,X_n]$ ...
0
votes
0answers
6 views

Is my proof showing that if products of powers of elementary symmetric polynomials are the same, then the power is same, correct?

Below proof is assuming this lemma, which can be proven easily: Let $\leq$ be a monomial ordering on $R[X_1,...,X_n]$. Let $f,g\in R[X_1,...,X_n]$ be nonzero polynomials such that $LC(f)$ is ...
0
votes
0answers
14 views

Does the fundamental theorem of symmetric polynomials hold in any ring?

Fundamental theorem of Symmetric polynomials: Let $R$ be a commutative ring and $e_0,...,e_n$ be the elementary symmetric polynomials of $R[X_1,...,X_n]$. Let $\Phi:R[X_1,...,X_n]\rightarrow ...
1
vote
1answer
26 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
0
votes
0answers
16 views

Is there a natural link between symmetric polynomials and symmetric algebra?

Let $R$ be a commutative ring and $R[X_1,...,X_n]^{S_n}$ be the ring of symmetric polynomials. I have learned some basic properties of this ring and the results are really similar to those by ...
2
votes
0answers
33 views

Looking for the name of particular collection of polynomials

I came across the following algebraic structure when working on a seemingly unrelated problem and am unable to find a name for it. Let $R$ be a commutative ring with identity. Given ...
5
votes
1answer
121 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
3
votes
1answer
65 views

Inequality of elementary symmetric polynomials

Let $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ with $\lambda_i>0$ for $i=1,2,3,4$. Let $$\sigma_k(\lambda)=\sum_{1\leq ...
2
votes
2answers
57 views

Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$

Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$ Suppose $a, b, c$ are roots of $P(x)$. $$P(x) = k(x - a)(x - b)(x - c)$$ But then I get $(k = 1)$ $$P(x) = x^3 - 12x^2 + ...
4
votes
1answer
34 views

Symmetric polynomials and g non symmetric

If $g=x_1+2x_2+3x_3, s_1=x_1+x_2+x_2, s_2=x_1x_2+x_1x_3+x_2x_3$ and $s_3=x_1x_2x_3$ , write $x_1, x_2$ and $x_3$ in function of $g, s_1, s_2$ and $s_3$.
2
votes
1answer
54 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
1
vote
0answers
62 views

System of (non linear) equations

Let $n \geq 2$. Could it be proved that the following system, with $z_k\in \mathbb C$, $ \begin{cases} z_1^n + z_{n}z_1^{n-1} + z_{n-1}z_1^{n-2} + \cdots + z_2z_1+z_1 & = 0 \\ z_2^n + ...
6
votes
4answers
223 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
0
votes
1answer
30 views

Algorithm for writing a symmetric polynomial P(x,y) in terms of x+y and xy

Assume I am given a polynomial of two variables by the list of its coefficients, $P(X,Y)=\sum_{i,j}C_{ij}X^i Y^j$, and that this polynomial is symmetric, i.e. $C_{ij}=C_{ji}$. I am looking for an ...
0
votes
0answers
27 views

Is every polynomial of two variables separable into a symmetric and antisymmetric part?

How can I split a polynomial into parts which are symmetric and antisymmetric under exchange of the variables? I have an explicit polynomial, which is a function of two variables (and some further ...
0
votes
1answer
29 views

Splitting a polynomial into parts which are symmetric and antisymmetric under exchange of variables.

How can I split a polynomial into parts which are symmetric and antisymmetric under exchange of the variables? I have an explicit polynomial, which is a function of three variables (and some further ...
1
vote
1answer
28 views

Splitting a polynomial into a symmetric and an antisymmetric part.

How can I split a polynomial into a symmetric and an antisymmetric part? I have an explicit polynomial, which is a function of three variables (and some further constants). The symmetry properties ...
0
votes
0answers
47 views

fundamental theorem of symmetric polynomials

Let $P_n(a,b,c)$ be a polynomial of variables $a,b,c$. By Newton's fundamental theorem of symmetric polynomials, there is a unique $P_n$ such that $$ x^n+y^n+z^n=P_n(x+y+z, x^2+y^2+z^2,x^3+y^3+z^3). ...
9
votes
1answer
140 views

On an identity with $n$ variables

We know the followings : $$\color{red}{{x_1}^2+{x_2}^2-2x_1x_2}=\color{blue}{(x_1-x_2)^2}$$ ...
1
vote
1answer
50 views

What is a symmetric polynomial?

I'm reading about symmetric polynomials at the moment, and came upon this statement: Let $G(x_1, \ldots, x_n)$ be a symmetric polynomial. Separate out $x_n$, representing $G$ as follows: $G_\nu ...
0
votes
1answer
63 views

How to solve the system of 4 equations of four unknowns

Solve this system of the four equations of four unknowns $a, b, c, d>0 $ $$ 165(a+b+c)=abc\tag1 $$ $$220(a+b+d)=abd \tag2 $$ $$297(a+c+d)=acd\tag3 $$ $$540(b+c+d)=bcd \tag4 $$ I tried to ...
1
vote
1answer
34 views

Newton polynomials

Consider the family of symmetric polynomials $\sum^n_{i=1} x_i^k\in\mathbf{Z}[x_1,\ldots,x_n]$. By the fundamental theorem on symmetric polynomials there is a unique Newton poylnomial ...
2
votes
0answers
33 views

Find the coefficients in $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$

Use evaluation homomorphisms $F[x_1,x_2, \dots, x_n] \to F$ to obtain the coefficients in: $$(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$$ where the $s_i$ ...
2
votes
3answers
67 views

Solving $n$ unknowns with $n$ independent equations

Is it always possible to solve $n$ independent equations with $n$ unknowns? Or is it possible to solve the following 3 equations with 3 unknowns? $$x + y + z = a$$ $$x^2 + y^2 + z^2 = b$$ $$x^3 + ...
1
vote
1answer
39 views

Why is the inverse of this matrix expressible in terms of elementary symmetric polynomials?

Suppose we have $N$ distinct real numbers $x_1, x_2, \ldots, x_N$ and consider the $N \times N$ matrix $$P = \left[\! \begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1 \\ x_1 & x_2 & ...
0
votes
0answers
17 views

counting how many entries from a given Young tableau contribute to hook length made from two differtn YTs

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ and define for $s=(i,j)\in R$ (it makes also sense for $s$ outside ...
1
vote
1answer
23 views

A question on a proof concerning resultants I don't quite get

http://planetmath.org/proofthatsylvestersmatrixequalstheresultant Heres a link to a proof I found concerning the relation between Sylvesters matrix and resultants. Most of it makes sense. I do have ...
3
votes
2answers
71 views

System of equations with 2 parameters

I have no idea how even to start! \begin{align*} (u^2+v^2)(u+v)&=15uv \\ (u^4+v^4)(u^2+v^2)&=85u^2v^2 \end{align*}
2
votes
1answer
53 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
1
vote
1answer
35 views

A system of non-linear equations with a small parameter

Is there any way to solve analytically the following system of equations to the leading order in $\epsilon$: $$\left\{ \begin{array}{rcl} \mu^2 \phi_1 + \lambda \phi_1 (\phi_1^2 + \phi_2^2) + ...
3
votes
1answer
55 views

(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define ...
3
votes
0answers
58 views

Is there an expression for Jack polynomials in terms of the power sum basis?

The Jack polynomials are the 1-parameter family of eigenfunctions of the differential operator: $$ D_\alpha = \frac{\alpha}{2} \sum_{i} x_i^2 \frac{\partial^2}{\partial x_i^2} + \sum_{i \neq j} ...
4
votes
2answers
54 views

$\{p_i\}$ generate the $k$-algebra of symmetric polynomials in $k[t_1, \dots, t_n]$ and are algebraically independent over $k$

Let $k$ be a field of characteristic $0$. For $j \ge 0$, let $p_j = t_1^j + \dots + t_n^j \in k[t_1, \dots, t_n]$. Prove that $p_1, \dots, p_n$ generate the $k$-algebra of symmetric polynomials in ...
10
votes
2answers
166 views

Show that the roots of the polynomial $x^4 - px^3 + qx^2 - pqx + 1 = 0$ satisfy a certain relationship

Here is the question: If the roots of the equation $$ x^4 - px^3 + qx^2 - pqx + 1 = 0 $$ are $\alpha, \beta, \gamma,$ and $\delta$, show that $$ (\alpha + \beta + \gamma)(\alpha + \beta + ...
8
votes
1answer
136 views

A generalization of arithmetic and geometric means using elementary symmetric polynomials

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$ P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) $$ then: The arithmetic mean of $a_1, \ldots, ...
2
votes
0answers
28 views

Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= ...
1
vote
5answers
47 views

Solving system when terms have both variables

$$x^3-3y^2x=-1$$ $$3yx^2 -y^3=1$$ This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was ...
1
vote
1answer
39 views

Find the equation which has key root $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$

In my last question which was Proving $x=\sqrt{a}+\sqrt{b}$ is the key root to solve $x^4-2(a+b)x^2+(a-b)^2=0$ ,I could find the coefficients(were very easy) of fourth-degree equation, so I went to ...
0
votes
1answer
85 views

How can I convince students a certain polynomial equation is symmetric?

How can I convince students that $p(x)=0$ is a symmetric equation if they ask me, where $p(x)$ is polynomial of degree $n$ with reals coefficients. For example : $A(x)=2x^4-9x^3+8x^2-9x+2=0 $ is ...
3
votes
2answers
46 views

Prove coefficients of polynomial are elementary symmetric polynomials

I want to show that for the $k$-th elementary symmetric polynomial $s_k:=\sum_{i_1\lt\cdots\lt i_k}X_{i_1}\cdots X_{i_k}\in R[X_1,\ldots,X_n]$ a monic polynomial that factors $\prod_{i=1}^n ...
3
votes
0answers
40 views

Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over ...
6
votes
1answer
84 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
0
votes
0answers
25 views

Finding a generalized form for taking the n$^{th}$ derivative of a falling factorial

I would like to make $$ \frac{d^n}{dx^n}[(x)_c] = n! \times e_{c-n}(x,x-1,\cdots,x-c+1) $$ Where $e_{c-n}(x,x-1,x-2,\cdots,x-c+1)$ is the elementary symmetric polynomial function But lets say that ...
2
votes
1answer
80 views

Irreducibility over $\mathbb{C}$ of symmetric polynomials

Problem. Find all elementary symmetric polynomials that are irreducible over $\mathbb{C}$. My attempt. It's easy to see that if we have polynomial $f(x_1, \dots, x_n)$ and it can be reduced to ...
2
votes
0answers
44 views

multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
0
votes
1answer
54 views

Solving a simple systems of equations

Update: 1) As @Amzoti mentioned, I made a mistake in the mathematica code. There should be spaces between x, y and z. So now the following code works: ...
1
vote
1answer
45 views

Representing a symmetric function in elementary symmetric functions

I'm trying to represent the following in the elementary symmetric functions base: $ \sum\limits_{i \neq j} x^2_i x_j $ and $ \sum\limits_{i \neq j} x_i^2 x_j^2 $ I don't really know how to ...
2
votes
1answer
63 views

Write the determinant as a polynomial expression in the elementary symmetric polynomials

How to write $\det\begin{bmatrix}x_1&x_2&x_3&x_4\\x_2&x_3&x_4&x_1\\x_3&x_4&x_1&x_2\\x_4&x_1&x_2&x_3 \end{bmatrix}$in terms of elementary symmetric ...
1
vote
1answer
24 views

Polynom as sum/product of symmetric polynoms

I have a polynom $(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1)$ and I need to express as sum/product of elemental symmetric polynoms $s_1,s_2,s_3$. I know there is an algoritm for ...
0
votes
1answer
39 views

Build a polynomial

I have $f=x^3 + ax^2 +bx +c \in \mathbb C[x], \alpha_1,\alpha_2,\alpha_3 \in \mathbb C$ are roots of $f$. $\beta_1 = {\alpha_1 \over \alpha_2} + {\alpha_2 \over \alpha_3} + {\alpha_3 \over \alpha_1}, ...