# Tagged Questions

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

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### When is a recurrence the sum of the powers of the roots of a polynomial?

Newton's formula allows one to calculate the sum $S_n(P)$ of the $n$th powers of the roots of a given monic polynomial $P$ without finding the roots explicitly. (This works even when the roots ...
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### How to solve this equation algebraically [closed]

Solve the following simultaneous equations on the set of real numbers: \begin{cases}x^2 + y^3 = x+1 \\ x^3+y^2=y+1\end{cases} Thanks for helping!
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### What are the one-dimensional elements in the ring of symmetric functions?

The verification principle for $\lambda$-rings says (if I'm understanding correctly) that if you have a $\lambda$-ring $A$, and an equation using only $\lambda$-ring operations (addition, ...
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### Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
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### Solving system of non-linear equations.

So I'm trying to find the stationary points for $$f(x,y,z) = 4x^2 + y^2 +2z^2 -8xyz$$ Setting the partial derivatives to zero leads to: $$x-yz=0 \\ y-4xz=0\\z-2xy=0$$ Substiting $z=2xy$ into the ...
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### Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
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### Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
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### Schur polynomials with variables raised to a fixed power

For a partition $\lambda$ let $s_{\lambda}(x_1,\dots,x_k)$ denote the Schur polynomial in $k$ variables associated to $\lambda$ (let's assume that $k$ is sufficiently large compared to $\lambda$ that ...
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### A binomial symmetric sum

Denote \begin{align*} \text{Sym}_k(\textbf{x})=\sum_{i_1<\cdots<i_k}x_{i_1}\cdots x_{i_k} \end{align*} as the $k$th elementary symmetric sum in monomials $\textbf{x} = (x_1, \cdots, x_n)$. If ...
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### Factorize $(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$

I am unable to factorize this over $\mathbb{Z}:$ $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$$ Since, this from an exercise of a book (E. J. Barbeau, polynomials) it must have a neat ...
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### if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$ and $A+B+C = 0$ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$$ and $$A+B+C = 0$$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
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### Efficient way to compute the symmetric reduction of special polynomials (specially for resolvents)

By the Fundamental Theory of Symmetric Polynomials every symmetric polynomial in $K[x_1, \dots, x_n]$ can be written uniquely in the elementary symmetric functions $s_1, \dots, s_n$. I know there are ...
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### Is there any Newton type identity for the following ordered matrix symmetric polynomial?

I have the following sum of matrices $A_i,\ i=1,2,\cdots,\ n$ $$\sum_{n\ge i_1>i_2>\cdots>i_k\ge 1}A_{i_1}A_{i_2}\cdots A_{i_k}$$ This looks like an elementary symmetric polynomial but it is ...
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The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by $$\bigg(\sum_{k=0}^nx^k\bigg)^p~=~\frac1{(p-1)!}~\sum_{k=0}^{np}\frac{(n-|n-k|+p-1)!}{(n-|n-k|)... 1answer 28 views ### Symmetric Polynomial in roots is in F[X] I recently came across the following claim. Let F be a field of characteristic 0. Let f\in F[X] have roots y_1, \ldots , y_d in the algebraic closure of F. Define$$ g_h = \prod_{1\le \...
Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}}$ ...