Question

Help with elementary symmetric polynomials problem!

Answer 1

\[x _{1}^{3} + x _{2}^{3} + x _{3}^{3} + x _{4}^{3}\]

Answer 2

Decompose this polynomial into elementary symmetric polynomials

Answer 3

Pick a pair of variabs. (and the other pair) and decompose both pairs as
\[a^3 +b^3 =(a+b)(a^2 - ab + b^2)\]

Answer 4

Soo @nazgul ?

Answer 5

is that it? the answers i found on the internet on different problems with elementary symmetrical polynomials seem to be more complicated

Answer 6

\[ \text{ that means} (a+b)([a+b]^2 - 3ab) \\ \text{you are expected to express everything as combinations of}\\
\quad C_k*(x_1 + x_2 + x_3 + x_4)^k \quad \text{and\or} \\ \,\,\,B_n*(x_1x_2x_3x_4)^n\]

Answer 7

Of course the base-station would be \[ (x_1 + x_2 + x_3 + x_4)^4 = ..\]

Answer 8

then your task is reduced to again decomposing
\[ \Sigma_{j\neq k} \,\,x_j^3*x_k \]

Answer 9

Which is done by \[ \Sigma_{j\neq k} \,\,x_j^3*x_k \,\, + \Sigma_{j\neq k ,\\ j\neq n} x_j^2x_kx_n =\Sigma(x_jx_k)(x_1^2 + x_2^2 + x_3^2 + x_4^2 )\]

Answer 10

It is a long tedious "accounting" procedure

Answer 11

yea thanks , I think I got it now , you are amazing ;)

Answer 12

BTW \[ (x_1^2 + x_2^2 + x_3^2 + x_4^2 )\] is easily decomposable