-- -*- M2-comint -*- {* hash: 369610110 *}

i1 : R = QQ[a..d];

i2 : degrees R

o2 = {{1}, {1}, {1}, {1}}

o2 : List

i3 : heft R

o3 = {1}

o3 : List

i4 : S = QQ[a..d,DegreeRank => 4];

i5 : degrees S

o5 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}

o5 : List

i6 : heft S

o6 = {1, 1, 1, 1}

o6 : List

i7 : T = QQ[a,b,Degrees => {1,-1}]

o7 = T

o7 : PolynomialRing

i8 : degrees T

o8 = {{1}, {-1}}

o8 : List

i9 : heft T

i10 : U = QQ[a..d,Degrees => {{2,0},{1,-1},{0,-2},{-1,-3}}]

o10 = U

o10 : PolynomialRing

i11 : degrees U

o11 = {{2, 0}, {1, -1}, {0, -2}, {-1, -3}}

o11 : List

i12 : heft U

o12 = {3, -2}

o12 : List

i13 : hilbertSeries U

                          1
o13 = ----------------------------------------
            2         -1       -2       -1 -3
      (1 - T )(1 - T T  )(1 - T  )(1 - T  T  )
            0       0 1        1        0  1

o13 : Expression of class Divide

i14 : describe ring numerator oo

o14 = ZZ[T , T , Degrees => {3, -2}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
          0   1                                       {Weights => {-3, 2}}
                                                      {GroupLex => 2     }
                                                      {Position => Up    }

i15 : hilbertSeries(U,Order => 8)

           -1 -3    -2      -1    2    -2 -6    -1 -5
o15 = 1 + T  T   + T   + T T   + T  + T  T   + T  T
           0  1     1     0 1     0    0  1     0  1

o15 : ZZ[T , T ]
          0   1

i16 : R = QQ[x,y,Heft=>{3}];

i17 : degree R

      1
o17 = -
      9

o17 : QQ

i18 :