Question, part (a):  If

    y1 = x1^2 + x2^2
    y2 = x1^3 + x2^3
    y3 = x1^4 + x2^4

can you then compute a polynomial relation between y1,y2,y3?
Think of the above as a polynomial map:
   (x1,x2)  -->  (y1,y2,y3)
and this parametrizes a 2-dimensional surface in a 3-dimensional
space, and now you have to find an equation for that surface.

Part (b):

for k to 8 do
    f[k] := x1^k + x2^k
od;

Can you express f[5] as a polynomial in y1,y2,y3?
If so, then give that polynomial.
Same question for (x1*x2)^2.

For which k's (k=1..8) can you express f[k] as a
polynomial in y1,y2,y3?  (don't need to give the polynomial).