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

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9
votes
1answer
206 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 ...
5
votes
4answers
977 views

Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?

Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
11
votes
1answer
2k 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 ...
3
votes
3answers
201 views

Do these special power functions generate all homogeneous symmetric polynomials?

Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree. My question is as follows. To be succinct, let's say we have four ...
4
votes
2answers
353 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!
5
votes
0answers
115 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 $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ ...
4
votes
2answers
1k 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 ...
6
votes
3answers
659 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 ...
2
votes
1answer
163 views

An identity on symmetric polynomial

In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set ...
6
votes
2answers
274 views

Number of terms in a monomial symmetric polynomial

Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...
5
votes
2answers
1k 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 ...
15
votes
2answers
2k 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 ...