--==================================================================--
-- Tutorial 5: Monomial Orders and Groebner Bases                   --
--==================================================================--

-- Use C-x 2 to get a horizontally split window or 
-- Use C-x 3 to get a vertically split window.

-- Select the lower (or right) window and Hit F12 to start M2.

-- Hit F11 to send code from this screen to M2.

-- for help with M2 type (works in version 1.1)

viewHelp

----------------------------------------------------------------------
-- Monomial Orderings                                               --
----------------------------------------------------------------------

-- Lex Orderings -----------------------------------------------------

-- Consider the ring:

R = ZZ/2[x_1..x_4];

-- Give the above ring the Lex ordering with 

-- x_1 > x_2 > x_3 > x_4.  

-- Show off that you have the correct ordering with some examples of
-- polynomials with ordered terms.  (When a ring has a specified
-- monomial ordering, Macaulay 2 will always use that ordering when
-- returning polynomials)



-- Now keep the ring above, but give it the Lex ordering with 

--x_4 > x_3 > x_2 > x_1



-- GLex (GrLex) Orderings --------------------------------------------

-- Consider

R = ZZ/2[x_1..x_4];

-- Give the above ring the GLex ordering with 

-- x_1 > x_2 > x_3 > x_4.  

-- Show off that you have the correct ordering with some examples of
-- polynomials with ordered terms.



-- GRevLex Orderings -------------------------------------------------

-- The default monomial ordering in Macaulay 2 is the GRevLex
-- ordering.

R = ZZ/2[x_1..x_4];

-- Show off the GRevLex ordering of the above ring with some examples.



-- Comparative Ring Theory -------------------------------------------

-- Now take the following three rings:

A = ZZ/2[x_1..x_4] -- Equip this ring with the Lex ordering
B = ZZ/2[y_1..y_4] -- Equip this ring with the GLex ordering
C = ZZ/2[z_1..z_4] -- Let this ring have the GRevLex ordering

-- Find a f(t_1..t_4) such that the lead terms of 
-- f(x_1..x_4), f(y_1..y_4), and f(z_1..z_4)
-- are all different. 



-- Product Orderings -------------------------------------------------

-- A special type of ordering is called a Product ordering.  Here are
-- two rings.  The second one has a Product ordering on it.  Try some
-- "test" polynomials to see if you can figure out what the Product
-- ordering is doing.  When you get done, confer with your neighbor
-- and see if you can come to an agreement.

A = ZZ/2[x_1..x_4]
B = ZZ/2[y_1..y_4,MonomialOrder=>{1,3}]



-- What about this Product ordering?

A = ZZ/2[x_1..x_4]
B = ZZ/2[y_1..y_4,MonomialOrder=>{2,2}]



-- What about this Product ordering?

A = ZZ/2[x_1..x_4]
B = ZZ/2[y_1..y_4,MonomialOrder=>{1,1,1,1}]



----------------------------------------------------------------------
-- Groebner Bases                                                   --
----------------------------------------------------------------------

-- You compute Groebner bases in Macaulay 2 using the command "gb"

A = ZZ/2[x_1..x_4]
I = ideal (x_1^3-x_2*x_3^4,x_1*x_2*x_3)
gb I

-- OK, maybe that doesn't tell you much.  To get the generators of a
-- gb use "gens"

gens gb I

-- OK, that gives you a matrix.  Though it is a little hard to see
-- what the generators of the Groebner basis are.  Use transpose to
-- make this more clear.



-- When computing Groebner bases, the monomial order matters a lot.
-- Consider the following ring:

n = 4;
A = ZZ/2[a_1..a_(2*n),b_1..b_(4*(n-1)),c_1..c_(4*(n-2)),d_1..d_(4*(n-3))]

-- Now construct two matrices, X and Y:
X = matrix{{a_1,b_1,b_5,b_9},{b_3,a_3,c_1,c_5},{b_7,c_3,a_5,d_1},{b_11,c_7,d_3,a_7}}
Y = matrix{{a_2,b_2,b_6,b_10},{b_4,a_4,c_2,c_6},{b_8,c_4,a_6,d_2},{b_12,c_8,d_4,a_8}}

-- Consider the following ideal, which is generated by the entries of
-- the matrix X*Y-Y*X:

I = ideal(X*Y-Y*X);

-- Try to compute a Groebner basis of I:



-- When you think Macaulay 2 has had enough, kill Macaulay 2.  Try to
-- do it without closing this window.

-- Now consider the following ring with the following Product order:

n = 4;
A = ZZ/2[a_1..a_(2*n),b_1..b_(4*(n-1)),c_1..c_(4*(n-2)),d_1..d_(4*(n-3)),MonomialOrder=>{8,12,8,4}]

-- Again use these matrices:

X = matrix{{a_1,b_1,b_5,b_9},{b_3,a_3,c_1,c_5},{b_7,c_3,a_5,d_1},{b_11,c_7,d_3,a_7}}
Y = matrix{{a_2,b_2,b_6,b_10},{b_4,a_4,c_2,c_6},{b_8,c_4,a_6,d_2},{b_12,c_8,d_4,a_8}}

-- Can you describe a relationship between the matrices above and the
-- Product order chosen?



-- Now take the same ideal:

I = ideal(X*Y-Y*X);

-- Compute a Groebner basis of I.  Wait for it...



-- If you think this is neat, check out:

-- F. Hreinsdottir "A Case Where Choosing a Product Order Makes the
-- Calculation of a Groebner Basis Much Faster" Journal of Symbolic
-- Computation (1994) 18, 373-378.

-- and 

-- F. Hreinsdottir "An improved term ordering for the ideal of
-- commuting matrices" Journal of Symbolic Computation (2006) 41,
-- 999-1003.

----------------------------------------------------------------------
-- Counting Monomials                                               --
----------------------------------------------------------------------

-- Start by restarting!

restart

-- In the paper we consider the ring: R = k[x_1, x_2, x_1^(-1), x_2^(-1)].
-- Set k = ZZ/2 and build R in Macaulay 2.

R = 

-- Now build S as in the paper.

S = 


-- As a gesture of friendship, we've typed in the set M for you:

M = {x_1^2*x_2, x_1*x_2^2, x_1^(-1)*x_2^2, x_1^(-2)*x_2, x_1^(-2)*x_2^(-1), x_1^(-1)*x_2^(-2), x_1*x_2^(-2), x_1^2*x_2^(-1)}

-- Use Macaulay 2 to construct the map Psi : S -> R mapping the y's to
-- the elements of M

Psi = 

-- Now compute the kernel of Psi, call it k

k = 

-- At this point, we need to know if k is a binomial ideal. Since k
-- has many elements, we should use a routine to do it for us.

isBinomial = f -> # terms f <= 2;
isBinomialIdeal = I -> (
     L := flatten entries gens gb I;
     i := 0;
     while isBinomial L_i and i + 1 < #L do if isBinomial L_i then (i  = i + 1) else break;
     isBinomial L_i);

-- Check it out, you should get "true" out.

isBinomialIdeal(k)

-- Now we have a somewhat complex routine.  Someone has commented the
-- first three lines of the routine.  Read through the routine and
-- comment the last 4 lines.

homogeneousPart = I -> (
     A := ring I; -- allows us to work with the ring of I
     t :=local t; -- gives us a "safe" variable to use
     B := coefficientRing A [t_0..t_(# gens A),MonomialOrder=> Eliminate 1,Degrees => prepend({1},degrees A)]; -- gives us a ring to eliminate with     
     i := map(B,A,{t_1..t_(# gens A)}); 
     Ih := homogenize(i I,t_0); 
     j := map(A,B,flatten{0,gens A}); 
     j ideal selectInSubring(1, gens gb Ih) 
     );

-- Now we can continue to follow the paper with:

H = homogeneousPart k

-- Now compute the reduced Hilbert series of H, the Hilbert polynomial
-- of H, and the Hilbert function of H.  From this Hilbert function
-- you have your function "f" from the paper.



-- To finish, all you need to do is compute the ideal of lead terms of
-- k.  Call it ink:

ink = 

-- Now compute the reduced Hilbert series of ink, the Hilbert
-- polynomial of ink, and the Hilbert function of ink.  From this
-- Hilbert function, you have the function "g" from the paper.



--==================================================================--
-- FIN                                                              --
--==================================================================--