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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.

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Some lemma on elementary symmetric polynomial

$$e_t(x_2,\ldots,x_n) = \sum^{n-t}_{i=1} (-1)^{i+1} \frac{e_{t+i}(x_1,\ldots,x_n)}{x^i_1}\text{ for every } 0\leq t < n.$$ $e_t$ is the $t^{th}$ elementary symmetric polynomial in the variabel $\{x_2\ldots x_n\}$ $e_{t+i}$ is the ${(t+i)}^{th}$…
laodeam
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Solving $\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$ with symmetric polynomials

I need to solve the following with symmetric polynomials, but I do not understand what to do here. $$\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$$
padalad499
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An identity for the sum of squared complete homogeneous symmetric polynomials in terms of power sums

Consider $h_k(x)$ the complete homogeneous symmetric polynomials and $p_k(x)$ the power sum polynomials for some set of variables $x$. If there a finite number of non-zero $p_k$, is there an identity that expresses $\sum_{k=0}^\infty h_k^2$ in terms…
WLV
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The square root of the discriminant function

Define the discriminant by $$\Delta^2 = \prod_{i > j}(x_i -x_j)^2$$ Clearly $\Delta^2$ is a symmetric polynomial. My question is, why is the square root of of $\Delta^2$, i.e $$\Delta = \prod_{i > j}(x_i -x_j)$$ Not symmetric?
user486995
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Principal specialisation on the forgotten symmetric polynomial looks exceptional, why?

for any symmetric polynomial $u(\lambda,n)$ the principal specialisation is usually defined as $ps(u(x_1,x_2,x_3, ..)) = u(1,q,q^2,q^3, ..)$. See also https://www.math.upenn.edu/~peal/polynomials/standardSymmetricFunctions.htm. I was surprised to…
Wouter M.
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How is this power sum equation related to the Newton-Girard identity?

I am trying to understand the logic behind solving this problem from youtube. Given the equation: $p_n-e_1p_{n-1}+\dots+n(-1)^ne_n = 0$ And the Newton-Girard identity: $ke_k = e_{k-1}p_1-e_{k-2}p_2+\dots+(-1)^{k-1}e_0p_k$ How are these 2 equations…
ng.newbie
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Choose the variables so that the weighted symmetric polynomial is minimal.

I've been struggling with the following problem for hours: Consider the expression $p^2\frac{x}{y+z}+q^2\frac{y}{x+z}+r^2\frac{z}{x+y}$, where $p,q,r>0$ are parameters. Choose $x,y,z\ge0$ so that the value of the expression is a minimum. Any ideas…
Ralph
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Maximum value of symmetric elementary polynomials?

Assume a function $f(a_1,a_2,...,a_t)$ over the $t$ variables $a_i$ (each $a_i$ is $N$-bit). I read that if for such functions it holds that $|f(a_1,a_2,...,a_t)|
Jimakos
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What is a similar relation for $\sum_{n=0}^\infty{\frac{B_n(0!F(s),1!F(2s),\dots,(n-1)!F(ns))}{n!}}$

A result of the exponential formula is $$\exp{\left(\sum_{n=1}^\infty{\frac{a_n}{n!}x^n}\right)}=\sum_{n=0}^\infty{\frac{B_n(a_1,\dots,a_n)}{n!}x^n}$$ Now define $$F(x)=\sum_{n=1}^\infty\frac{1}{(t_n)^x}$$ for some set of real numbers…
tyobrien
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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 &= x_1x_2 + x_1 x_3 + \cdots + x_{n-2}x_n+ x_{n-1}x_n…
Human
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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 constants). The symmetry properties should be with…
Thomas
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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 should be with respect to all three variables.
Thomas
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Perform $x_1^3+x_2^3+\cdots+x_n^3$ by basic symetric polynomials.

Perform $x_1^3+x_2^3+\cdots+x_n^3$ by basic symetric polynomials $\sigma_1,...,\sigma_n$. $\sigma_1=x_1+x_2+\cdots x_n\\ \sigma_2=\sum\limits_{i
user41499
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Relation between two polynomial subrings

Let $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ be ring in $x_1,\ldots, x_n$ with coefficients from $\mathbb{Z}$. Let $e_1=x_1+x_2+\cdots + x_n$, $e_2=\sum_{i
Maths Rahul
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Express $x^2 y^2 + y^2 z^2 + z^2 x^2$ in the form of $x+y+z$, $xy+yz+zx$ and $xyz$?

I'm not familiar with Symmetric polynomial. As title, is it possible to express $$x^2 y^2 + y^2 z^2 + z^2 x^2$$ using $x+y+z$, $xy+yz+zx$ and $xyz$? Is there a trick for such problems?
athos
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