# Questions tagged [symmetric-polynomials]

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

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### Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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### Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions

I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is \begin{align} a^{\...
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### How to show that $X_m$ is a zero of this polynomial in $R[X_1,…,X_m][X]$?

I posted this question a few days ago but the images didn’t work so nobody knew what I was talking about. I'm self-studying through Amann/Escher Analysis 1 and I'm stuck on a problem in I.8. Here's ...
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### Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$

Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}$ , $b=\sqrt{3}-\sqrt{5}+\sqrt{7}$, $c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate: $$\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$$ What I have tried so far is writing the ...
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### Root transformation to make normalized coefficient match

Let $f(x) = \prod_{i=0}^n(1+a_ix), a_i \neq 0$, and $C(f(x)): = (c_0, c_i, \cdots, c_n)$, where $f(x) = \sum_{i=0}^n c_i x^i$. For a given monomial $g(x) = (1+mx)(1+nx)$, i'm interested in a quantity ...
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### Reciprocal binomial coefficient polynomial evaluation

The conventional binomial coefficient can be obtained via $$f(x, n) = (1+x)^n = \sum_{i=0}^n { n \choose i} x^i$$ And the function $f$ can be every efficiently performed on evaluation. I'm ...
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### Is it possible to obtain roots from the weighted sum of other polynomials in their root form?

Given $n$ polynomials $A_1, A_2, .. A_n$with the same degree $M$. $A_i = \prod_{j=0}^M(1+Q_{ij}x)$. In their root form $Q$, $Q_{ij} \in \mathbb{R}$. And the function, I'm interested in, $B$ is a ...
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### How does the fundamental theorem of symmetric polynomials imply that this number is rational?

In the Problems from the Book by Titu Andreescu, there is a proof of Example 9 on page 494 with the following: Example 9. Let $f$ be a monic polynomial with integer coefficients and let $p$ be a ...
1 vote
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### Galois group of general equation of degree n

Let $K$ be a field and $\{t_1,t_2,\ldots,t_n\}$ be an algebraically independent set over $K$. For every permutation $\sigma \in S_n$ we have an automorphism of $K(t_1,t_2,\ldots,t_n)$ given by mapping ...
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### Definition of Macdonald polynomial $P_\lambda^{\mathfrak{g}}$ associated to a Lie algebra $\mathfrak{g}$ (unlike $P_\lambda$)

I want to find the definition for the Macdonald polynomial associated to a Lie algebra $\mathfrak{c}_n$, i.e. $P_\lambda^{\mathfrak{c}_n}(x,t,q)$. This appears in the physics paper https://arxiv.org/...
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### Local max of ratio of elementary symmetric polynomials

Let $e_k(x_1,\cdots, x_n) := \sum_{1\leq j_1 < \cdots < j_k \leq n}x_{j_1}\cdots x_{j_k}$ be elementary symmetric polynomials (see e.g. https://en.wikipedia.org/wiki/...
$x_i>0, \forall\ 1\leq i\leq n$ is equivalent to $\sigma_1,\cdots,\sigma_n$ are all $>0$, where $\sigma_i$ are the elementary symmetric polynomials. Here $\sigma_1=x_1+\cdots+x_n, \sigma_2=... 0 votes 0 answers 75 views ###$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$Problem: Prove that in any triangle ABC, the following inequality is true: $$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$$ Anyone can help me? I tried AM-GM for right ... 0 votes 0 answers 20 views ### Mathematica package for computing Macdonald polynomials 0 I want to implement computation of Macdonald polynomials in mathematica. A similar question was raised in another question 5 years ago (Macdonald-Koornwinder polynomials?), but received no clear ... 3 votes 1 answer 89 views ### Prove that:$2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$Problem: For$a,b,c\ge0: ab+bc+ca>0.$Prove that: $$2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$$ Recently, i have seen a post on AoPS link My approach: ... 2 votes 2 answers 134 views ### In triangle. Prove that:$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}\$
Problem: Given a,b,c are length of triangle. Prove that: $$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$$ Happy Vietnamese Women's ...