--  This is code related to HW 3 for Math 615, Winter 2013
--  By: Daniel Erman


--  Exercise 1
--  The whole point of Exercise 1 is to do it by hand!
--  But it might be helpful to use M2 to check that you got the right answers.

--  Here is how to solve the "original version" of the problem on M2, 
--  using things up with the lex term order.
S = QQ[x,y,z,w, MonomialOrder => Lex]
I = ideal(x*z-y^2, x*w-z^2, y*w-x*z)
f = x^2*y^2*w^2 - y^4*z^2
f % I
--  Since f % I does not equal 0, it follows that f does not lie in I.


--  Exercise 2
--  The syntax in M2 is pretty simple.  Here is an example that does not work.
f = x^2+y^2
g_1 = x^2
g_2 = y^2
(f % g_1) % g_2 == (f % g_2) % g_1


--  Exercise 3
--  I think that this problem is best solved by experimenting on M2.
--  It should be solvable with any term order, though you can see
viewHelp MonomialOrder
--  If you don't specify a term order, then M2 will choose GRevLex.
viewHelp GRevLex


--  Exercise 4
--  There are many ways to solve this problem, and solving it by hand would be useful.
--  But it might also be useful to compute a Grobner Basis.  Here is the syntax for
--  computing and viewing a grobner basis for a simple ideal.
S = QQ[x,y]
I = ideal(x^3,x^2*y+y^3)
G = gb I
--  Note that M2 suppresses the list of generators.  If you want to see the actual elements in the 
--  Grobner basis, try:
gens G
--  Or, if you want to realize this as a list you should do
flatten entries gens G
--  The "flatten" command drops a redundant set of parentheses.

--  Exercise 7
--  You're asked to do this by hand, but it is worth checking your answer in M2.
--  To get the lex order, use:
S = CC[x,y, MonomialOrder => Lex]
--  Of course, the computation would be the same if you replaced CC by QQ.