The symmetric-polynomials tag has no wiki summary.
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1answer
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Values of symmetric polynomials determining inputs
I've got a question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots \in ...
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0answers
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Different bases of symmetric polynomials
Let the symmetric group $S_n$ act on $R^n$ by permuting coordinates and consider the ring of symmetric polynoials. These symmetric polynomials can be written fro example as polynomials in the ...
3
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0answers
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Significance behind splitting of (x+y)^7-(x^7+y^7)? [closed]
A quick check via third roots of unity establishes$$(x+y)^7-(x^7+y^7) = 7xy(x+y)(x^2+xy+y^2)^2$$
Some questions:
What is the significance of this factorization?
Does it have any connection to the ...
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2answers
222 views
When powers of matrices are represented as a sum of integral matrices
There is a ring $R$ and its subring $K$ with unit. We have a matrix $A$ of order $n$ over $R$. Someone said, that if $A^m$ for $m=1,...,n$ can be represented as a sum of matrices over $R$ which a ...
4
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1answer
260 views
Normal forms for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$
Is there a standard normal form for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$? Or, put another way, is there a nice way to describe the orbit space of the natural (diagonal) action ...
4
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2answers
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Reference request on symmetric polynomials
A version of this question on stackexchange got a few comments from one person and no answers.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,\ldots,x_n$ (so in ...
4
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1answer
220 views
A nice generating set for the symmetric power of an algebra
I'm looking for a reference for the following fact.
Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ ...
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0answers
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Multivariate polynomial with positive coefficients
This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...
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0answers
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algebra of endomorphisms over the diagonal invariants
Let $k$ be a field of characteristic 0 (say $\mathbb{C}$).
Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$.
It is known that the ...
1
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1answer
215 views
Majorization of power sum symmetric functions
I'm wondering if there is a characterization for whether
$$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \mathbb{R}^+$}.$$
(Or ...
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2answers
486 views
Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
5
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2answers
712 views
Diagonal invariants of the symmetric group on $k[X_1,X_2,…,X_n,Y_1,Y_2,…,Y_n]$
This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
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0answers
235 views
Looking for product of symmetric polynomials evaluated at roots of unity
Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...
