-1
votes
1answer
121 views

Algorithm to find symmetric function given specialization

I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree ...
2
votes
1answer
96 views

Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...
4
votes
1answer
224 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 $S^n(A)$ ...
4
votes
1answer
239 views

To compute minors of Jacobian of symmetric polynomials

For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$ one has Jacobian, expressed by the $(n \times n)$-determinants: $$ ...
0
votes
0answers
243 views

Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$. Let $I=(p_1,p_2)$ be an Ideal of ...
0
votes
1answer
213 views

Regular sequence of power sum symmetric polynomials in polynomial ring.

Let $S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$. Question: To show $p_m,p_{2m}, \dots,p_{nm}$ forms a ...
7
votes
1answer
383 views

Characterizing intersection of zero sets of elementary symmetric polynomials on R^n

Stated simply, the question is: Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb ...
5
votes
1answer
627 views

Symmetric polynomials theorem

Hello all, I would appreciate comments on the following question: A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = ...