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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0answers
21 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 ...
6
votes
4answers
212 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', ...
1
vote
0answers
15 views

Symmetrize/antisymmetrize polynomial in three variables.

Splitting a polynomial of two variables into it's symmetric and antisymmetric part is done via $$ f(x,y)=f_s(x,y)+f_a(x,y)=\frac{1}{2}(f(x,y)+f(y,x))+\frac{1}{2}(f(x,y)-f(y,x)). $$ How do I split a ...
0
votes
1answer
24 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
18 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
23 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
26 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 ...
1
vote
0answers
39 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). ...
10
votes
1answer
131 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
44 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
62 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
32 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
32 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 $s_i$ ...
1
vote
3answers
63 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
37 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
14 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
69 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
46 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
31 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
0answers
40 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
56 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
49 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
155 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
133 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
37 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
84 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
43 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
38 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
81 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
77 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
41 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
52 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
40 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
58 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
23 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
38 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}, ...
0
votes
1answer
43 views

A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$ \frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1) $$ My first question is, how can I prove this? It seems trivial ...
4
votes
0answers
44 views

Question about primes of polynomial type.

It is well known that $50$ % of the primes are of the form $x^2 + y^2$. Many variants exists where a rational amount of primes is of some integer polynomial form. But I wonder ; are there integer ...
6
votes
2answers
59 views

$a_1^k+a_2^k+\ldots+a_n^k$ integer implies all integers?

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be rational numbers. Suppose that $a_1^k+a_2^k+\ldots+a_n^k$ is an integer for all positive integers $k$. Is it true that $a_1,a_2,\ldots,a_n$ ...
0
votes
0answers
23 views

Hilbert series computation for Hilbert scheme of $n$ points on $\mathbb C^2$

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{character}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $T$ acts on $x,y$ as $(t_1,t_2)$? ...
5
votes
4answers
395 views

Ring of polynomials as a module over symmetric polynomials

Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free ...
3
votes
0answers
47 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
4
votes
1answer
103 views

Solving systems of equations

I had a system of equations and i want know the perfect method to solve that: Solve for $X, Y, Z$ where : $\\$ $X^² = Y + a$ $Y^² = Z + a$ $Z^² = X + a$
0
votes
3answers
82 views

System of equations (contest problem)

Compute the ordered triple $(x,y,z)$ of positive real numbers that satisfies all three of the equations: $xy+x+y=19$ $yz+y+z=29$ $xz+x+z=53$ Please show me specific work and explain the law or ...
0
votes
1answer
47 views

Prove the following for integers

How can I show that ...
1
vote
0answers
43 views

Properties of the 'forgotten' symmetric polynomials

In I.G. Mcdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions 'f' are introduced very briefly as the result of applying an involution w to the monomial symmetric ...