with(Groebner):


# Question 1.
#------------
# Let I be the ideal I = (F) where F is:
F := {5*x^4*z+3*y^3*x^2, 5*y^4*z+3*x^3*y^2, x^5+y^5};


# Now consider the following polynomials in Q[x,y,z]
f1 := x;
f2 := y;
f3 := z;
f4 := (z+x)*(z+y);

# Is f1 in the radical ideal of I? If so, compute the smallest
# positive integer n for which f1^n is in the ideal I.
#
# Same question for f2, f3 and f4.


# Question 2.
#------------
#
# Let I be as in question 1.
# Compute generators g1..gk of the following ideal in Q[y,z]
#
#  I_yz := I intersected with Q[y,z]
#
# So: compute g1..gk such that I_yz = (g1..gk) = the ideal
#      generated by g1..gk.
# In other words: eliminate x from the equations in the set F.
#
# Looking at the ideal I_yz, can you guess what the radical ideal
# is?
#
# Same question for
#
#   I_xy := I intersected with Q[x,y].




# Question 3.
#------------

F := [x1+3-x2^2-x3^2,
      3+x1-x2^2+x2,
      -3+x1+x2+2*x1*x2-2*x2^3+2*x1*x3^2-2*x2^2*x3^2-x3^4-6*x3];



# Consider this as a map from C^3 to C^3.
# Input = 3 numbers x1,x2,x3.
# Output = 3 numbers y1=F[1], y2=F[2], y3=F[3].
# Is this map invertible, and if so, can you compute an inverse?
# That is, if y1=F[1], y2=F[2], y3=F[3], can you then express
# x1,x2,x3 as polynomials in y1,y2,y3?