**********************************************
** I do Hassett 2.9 (over Q) in Macaulay2. ***
**********************************************

i2 : R=QQ[x,y]

o2 = R

o2 : PolynomialRing

i3 : f=x^4+x^2*y^2+y^3-x^3

      4    2 2    3    3
o3 = x  + x y  - x  + y

o3 : R

i4 : I=ideal{f,diff(x,f),diff(y,f)}

             4    2 2    3    3    3       2     2    2      2
o4 = ideal (x  + x y  - x  + y , 4x  + 2x*y  - 3x , 2x y + 3y )

o4 : Ideal of R

i5 : hilbertSeries(coker gens I)

           2    4
     1 - 2T  + T
o5 = ------------
              2
       (1 - T)

o5 : Expression of class Divide

********************************************************************************
*So the hilbert series of Q[x,y]/in(I) is (1-T^2)^2/(1-T)^2=(1+T)^2=1+2T+T^2  ***
* In other words:                                                                *  
*(Q[x,y]/in(I))_0 has dim 1                                                      *
*(Q[x,y]/in(I))_1 has dim 2                                                      *
*(Q[x,y]/in(I))_2 has dim 1                                                     **
*(Q[x,y]/in(I))_i has dim 0 for i>0                                             **
* so dim Q[x,y]/in(I) = 4.    *****************************************************
**********************************************************************************







i7 :  gens gb I

o7 = | y2 x2 |

             1       2
o7 : Matrix R  <--- R

*******************************************************
** This is another way to calculate dim R/in(I)      **
** I asked Macaulay2 for the Groebner basis for I    **
   I see that the initial terms are y^2, x^2         **
   I conclude that a basis for R/in(I) is 1, x,y,xy  **
*******************************************************


***************************************************************
The next calculation shows that the normal form of x^5-y^5 in *
R/I is 0. So, x^5-y^5 is in I!                                *
***************************************************************
 
i10 : (x^5-y^5) % gb I                                      

o10 = 0

o10 : R

****************************************************************************************
Here is a second way to see that x^5-y^5 is in I. The first column                    **
shows how to write 320 (x^5-y^5) as a linear combination of the other three generators.**
****************************************************************************************

i11 : syz matrix {{f,diff(x,f),diff(y,f),x^5-y^5}}

o11 = {4} | 1280x2+1280xy+1280y2-5760x+960y+4320          480x2y+1120xy2+240x2-3240xy+840y2-540x+2520y+270             
      {3} | -320x3-320x2y-320xy2+1440x2-320xy-160y2-1440x -120x3y-280x2y2-60x3+840x2y-200xy2-140y3+150x2-840xy+30y2-90x
      {3} | -320x2y-320xy2-320y3+320x2+1920xy-320y2-1440y -120x2y2-280xy3+60x2y+1060xy2-280y3+60x2+180xy-840y2-90y     
      {5} | 320                                           0                                                            
      ---------------------------------------------------------------------------------------------------------------------------------------
      -32x2y+32xy2+144x2+216xy+24y2-1044x+72y+702     96x3+416x2y+448xy2-648x2-1272xy+336y2+648x+900y-162                
      8x3y-8x2y2-36x3-56x2y-4y3+270x2-24xy-42y2-234x  -24x4-104x3y-112x2y2+168x3+332x2y-108xy2-56y3-198x2-300xy+30y2+54x 
      8x2y2-8xy3-44x2y-44xy2-8y3+36x2+348xy-24y2-234y -24x3y-104x2y2-112xy3+24x3+260x2y+404xy2-112y3+12x2-216xy-300y2+54y
      0                                               0                                                                  
      ---------------------------------------------------------------------------------------------------------------------------------------
      32x3+288x2y-888x2+168xy+288x-36y+270             |
      -8x4-72x3y+224x3-28x2y-36xy2-126x2+12xy+6y2-90x  |
      -8x3y-72x2y2+8x3+292x2y-60xy2+36x2-96xy+12y2-90y |
      0                                                |

              4       5
o11 : Matrix R  <--- R

i12 :