Problem 1

Write a loop that expresses \(x^n+y^n+z^n\) in terms of elementary symmetric polynomials, for as many positive integers \(n\) as you can.

Solution

Following you can find a built in function in Mathematica:

xyzDecomp[n_] := SymmetricReduction[x^n + y^n + z^n, {x, y, z}][[1]]

an example where \(n=5\):

xyzDecomp[5]

and a plot where the \(x\) axis is \(n\) and the \(y\) axis is how long the computation takes in seconds: