Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [symmetric-polynomials]

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

A symmetric polynomial is a polynomial in several variables which is not changed after any permutation of variables. An important result about symmetric polynomials is the possibility to express any symmetric polynomial using elementary symmetric polynomials. Symmetric polynomials are useful, e.g., in connection with roots of polynomials (Vieta formulae). Another useful result concerns the Newton-Girard formulae, which expresses power sums in terms of elementary symmetric polynomials.

1334 questions
37
votes
2 answers

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 electrodynamics, we have Kirchhoffs law and so on, which result…
Nikolaj-K
  • 12,249
8
votes
1 answer

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} \frac{x_i}{x_i - x_j} \frac{\partial}{\partial…
Michael Cromer
  • 2,579
7
votes
2 answers

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 $$x + y + z$$
Andrew
  • 203
5
votes
1 answer

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 $\mathbb{Q}(x_1,...,x_n)$ naturally then the fixed field under this action is…
Dinesh
  • 1,737
5
votes
1 answer

Almost symmetric polynomial?

Let's say we have a polynomial $(x-y)(y-z)(x-z)$. This is not a symmetric polynomial, but it almost is. Every permutation of the variables results in a polynomial whose factors are multiples of the factors of the original polynomial. For example, if…
Exit path
  • 4,328
4
votes
4 answers

Sum of cubes of roots of a quartic equation

$x^4 - 5x^2 + 2x -1= 0$ What is the sum of cube of the roots of equation other than using substitution method? Is there any formula to find the sum of square of roots, sum of cube of roots, and sum of fourth power of roots for quartic equation?
Ghanaraj Devarajan
  • 43
3
votes
0 answers

A sequence of sym polynomials.

$\{f_i(x,y,z)\}$ is a sequence of polynomials with $f_0=1$. They satisfy $$f_{n+1}(x,y,z)=(x+z)(y+z)f_n(x,y,{z+1})-z^2f_n(x,y,z)$$ Pro. $f_i$s are symmetric. Easy observation I made as following: $f_{n+1}(x,y,z)=(x+z)(y+z)( f_n(x,y,z+1)-f_n(x,y,z) )…
Kirby Lee
  • 606
2
votes
2 answers

Find $x, y, z$ given $xyz = 60$, $x+y+z = 12$

If this can be solved in a way so that there is a unique value for $x$, $y$, and $z$, is there a name for the method that can be used to solve this problem? $$x+y+z = 12$$ $$xyz = 60$$ For this example, the values can be $3$,$4$,$5$, but I'm…
Robin Alvarenga
  • 123
2
votes
4 answers

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 symmetric polynomials. I was doing some explicit…
Daniel
  • 1,717
2
votes
1 answer

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: $G_\nu x_n^\nu + \cdots+ G_1 x_n + G_0$, where the $G_i$ are…
Jack M
  • 27,819
  • 7
  • 63
  • 129
1
vote
1 answer

Symmetric polynomials in some variables

Let $A_k$ be a set of polinomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ symmetric on $k
Silvio Sandro
  • 61
1
vote
1 answer

Symmetric polynomial and vieta's formulas

Can you help me solve this problem: Express the following symmetric rational function $$\frac{x_1}{-6x_2^2 - 7x_3x_2 - 6x_3^2} + \frac{x_3}{-6x_1^2 - 7x_2x_1 - 6x_2^2} + \frac{x_2}{-6x_1-7x_3x_1-6x_3^2}$$ as a function of $p$ and $q$, where $x_1$,…
Lyubomir Raykov
  • 45
1
vote
0 answers

Express polynomial in terms of elemetary symmetric functions

I'm working independently through the textbook Algebra by Cohn and have reached the section on elementary symmetric functions. I thought I had understood them but am confused by this question: Express $\Sigma x_1^3 x_2 x_3$ in terms of elementary…
Nathan
  • 11
1
vote
0 answers

Proving by induction a result relating to elementary symmetric functions

Let $f(x_1,...,x_n)$ be a symmetric polynomial, so that $f(x_{\sigma(1)},...,x_{\sigma(n)}) = f(x_1,...,x_n)$ for each $\sigma \in S_n$. For $1 \leq k \leq n$, we denote by $s_k$ the $k^{\text{th}}$ elementary symmetric function given by…
Menander I
  • 567
1
vote
0 answers

Clarification on proof including symmetric polynomials

This is regarding theorem 3 in this article. My problems begin after the equalities denoted by (5). My problems aren't so much about theory really, I think. I'm disregarding the authors' notation a bit and write…
Erik Vesterlund
  • 1,526
1
2 3