Question 1.
-----------

restart; with(Groebner):
F := [
   z^2-12*x*z-10*y*z+36*x^2+60*x*y+25*y^2,
   9*z^2+5*y*z-6*x*z+6*y^2+4*x*y+x^2-3*z-y+x,
   11*x*z-2*x*y-x^2+12*y*z+3*y^2-7*z^2,
   x^3+6*x^2*y-4*x^2*z+9*x*y^2-12*x*y* 
   z+4*z^2*x-2*x^2-x*y+8*x*z-9*z^2-6*y*z-y^2+3*z+y-x
];

Suppose that all equations in F are satisfied. What are then the
possible values for the variable x?
You're not allowed to use Maple's solve or gsolve.  What you are
allowed to do is to use an elimination ordering to get an equation
that only involves the variable x, and then factor that equation
with Maple's factor.


Question 2.
-----------

# Let I be the ideal I = (F) where F is:

restart; with(Groebner):
F := {y^5+2*x*y^2+2*x*y^3+x^2*y-4*x^3*y+2*x^5,
2*y^2+2*y^3+2*x*y-12*x^2*y+10*x^4, 5*y^4+4*x*y
+6*x*y^2+x^2-4*x^3};

Which of the following are in the ideal I?
f1 := x*y^2+3*y^3+x^2+4*x*y-x^3;
f2 := y^4+x^2+4*x*y;
f3 := y^4-x^2-4*x*y+4*x^3;
f4 := x*(y-1)*(x^2-x+1);

Of the ones that are not in the ideal, are there any that
are in the radical ideal? If so, which one(s)?


Question 3 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?

Question 3 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? (using the same
equations as in question 4a).
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).


Question 4.
-----------

Consider the ideals generated by:

I1 := {x^4 + y^4 + x^2*y^2, x^3 + y^5};
I2 := {x^5+y^6, x^6+y^7};

What is the intersection of those ideals?