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

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6
votes
2answers
2k views

symmetric polynomials and the Newton identities

I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
6
votes
3answers
1k views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
14
votes
1answer
4k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
11
votes
1answer
331 views

Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

I'm reading the introductory bits in Procesi's Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for ...
6
votes
4answers
751 views

Ring of polynomials as a module over symmetric polynomials

Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ a free ...
8
votes
4answers
199 views

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove ...
11
votes
4answers
1k views

Algorithm(s) for computing an elementary symmetric polynomial

I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing ...
6
votes
2answers
625 views

Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input

$\alpha _n ^n-1=0$ $\alpha _n=e^{2 \pi i/n}$ $$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$ I have read in Jim Brown's paper on page 5 that ...
5
votes
9answers
437 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
8
votes
2answers
343 views

Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?

User Eric Gregor and I were talking in chat and he mentioned this question and postulated the possibility of an approach through symmetric polynomials. After some thinking, I came to this: ...
1
vote
3answers
175 views

Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
16
votes
3answers
489 views

A generalized (MacLaurin's) average for functions

The average value of a function $y=f(x)$, on an interval $[a,b]$, is ${1\over {b-a}}\int_a^b f(t)dt$. This of course relates to the arithmetic average. It is easy to see that a corresponding formula ...
15
votes
2answers
4k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
9
votes
1answer
245 views

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
5
votes
1answer
147 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
9
votes
1answer
282 views

What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?

The product of monomial symmetric polynomials can be expressed as $m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$ for some constants $c_{\lambda\mu}^{\nu}$. In the case of Schur ...
6
votes
2answers
194 views

Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it

I got this question for homework and I've never seen anything similar to it. Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > ...
5
votes
0answers
137 views

Schur skew functions

Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ ...
5
votes
2answers
3k views

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete's formulas. I know that ...
4
votes
2answers
677 views

Need help solving a particular system of non-linear equations analytically

How would one go about analytically solving a system of non-linear equations of the form: $a + b + c = 4$ $a^2 + b^2 + c^2 = 6$ $a^3 + b^3 + c^3 = 10$ Thanks!
1
vote
0answers
73 views

Symmetrical Non-linear Constrained Equation

How would you approach this problem? I need all the possible solutions $$ (x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4) $$ which satisfy $$ ...
3
votes
2answers
72 views

Prove coefficients of polynomial are elementary symmetric polynomials

I want to show that for the $k$-th elementary symmetric polynomial $s_k:=\sum_{i_1\lt\cdots\lt i_k}X_{i_1}\cdots X_{i_k}\in R[X_1,\ldots,X_n]$ a monic polynomial that factors $\prod_{i=1}^n ...
2
votes
2answers
58 views

Proof for $A,B \in M_n(\mathbb{F})$ that if $[A,B]=tA$ for $0\neq t\in\mathbb{F}$, then $A^n=0$ [duplicate]

Statement. Suppose we have a square matrices $A,B$ of order $n$ over a field $\mathbb{F}$ of characteristics $0$ or $p>n$. If $[A,B]=AB-BA=tA$ for some nonzero $t\in\mathbb{F}$, then $A^n=0$. The ...
2
votes
1answer
303 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
1
vote
1answer
72 views

Proof that if $\mathrm{tr}\,A^k=0$ for all $k=1,\ldots, n$, then $A^n = 0$ [duplicate]

Statement. Suppose we have a square matrix $A$ of order $n$ over a field $\mathbb{F}$ of characteristics $0$ or $p>n$. There is a theorem that if $\mathrm{tr}\,A^k=0$ for all $k=1,\ldots, n$, then ...
1
vote
1answer
74 views

Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
1
vote
1answer
323 views

Decomposition of products of monomial symmetric polynomials into sums of them

I'm trying to make sense of the answer given in: this question I am stuck at the phrase 'where the partitions $\gamma$ result from adding, respectively, from $\alpha$ all distinct partitions obtained ...
1
vote
1answer
151 views

Equal-variables problem in three variables

This question resembles Vasile Cirtoajeā€™s equal-variables results, as explained here (although those results may be useless in the present problem). Let $s,p$ be two positive numbers with ...
0
votes
1answer
67 views

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 ...
0
votes
1answer
78 views

A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$ \frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1) $$ My first question is, how can I prove this? It seems trivial ...
0
votes
1answer
62 views

Factoring the group action on $\prod X_k^k $ gives another group?

Let's say you have a monomial symmetric polynomial, like the following $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 +X_2^1X_1^2X_3^3X_4^4 + \text{all permutations...} $$ Then you can ...