groebnerCone -- the cone whose interior weight vectors give the given initial ideal

i1 : R = ZZ/32003[symbol a..symbol d]

o1 = R

o1 : PolynomialRing

i2 : inL = {c^4, b*d^2, b*c, b^2*d, b^3}

       4     2        2    3
o2 = {c , b*d , b*c, b d, b }

o2 : List

i3 : L = {c^4-a*d^3, -c^3+b*d^2, b*c-a*d, -a*c^2+b^2*d, b^3-a^2*c}

       4      3     3      2                  2    2    3    2
o3 = {c  - a*d , - c  + b*d , b*c - a*d, - a*c  + b d, b  - a c}

o3 : List

i4 : weightVector(inL,L)

o4 = {8, 8, 3, 1}

o4 : List

i5 : groebnerCone(inL,L)

o5 = (| 0  0  |, | 1  0 |)
      | 0  0  |  | 0  1 |
      | -2 -3 |  | -2 3 |
      | -3 -4 |  | -3 4 |

o5 : Sequence

i6 : I = monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o6 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o6 : Ideal of R

i7 : time (inLs,Ls) = gfan I
LP algorithm being used: "cddgmp".
     -- used 0.002484 seconds

           2          2   2      3          2   2      3          2   2  
o7 = ({{b*d , a*d, a*c , a c}, {c , a*d, a*c , a c}, {c , b*c, a*c , a c,
     ------------------------------------------------------------------------
      3      3        4     2   2      3        3     2     3        2    3  
     a d}, {c , b*c, b , a*c , a c}, {c , b*c, b , a*c }, {c , b*c, b d, b },
     ------------------------------------------------------------------------
         2   2         2        2   2    3            2        2    3     3  
     {b*d , b d, a*d, a c}, {b*d , b d, b , a*d}, {b*d , b*c, b d, b , a*d },
     ------------------------------------------------------------------------
       4     2        2    3         3      2                  2    2      3
     {c , b*d , b*c, b d, b }}, {{- c  + b*d , - b*c + a*d, a*c  - b d, - b 
     ------------------------------------------------------------------------
        2      3      2                  2    2      3    2      3      2 
     + a c}, {c  - b*d , - b*c + a*d, a*c  - b d, - b  + a c}, {c  - b*d ,
     ------------------------------------------------------------------------
                   2    2      3    2      4    3      3      2            
     b*c - a*d, a*c  - b d, - b  + a c, - b  + a d}, {c  - b*d , b*c - a*d,
     ------------------------------------------------------------------------
      4    3      2    2      3    2      3      2              3    2      2
     b  - a d, a*c  - b d, - b  + a c}, {c  - b*d , b*c - a*d, b  - a c, a*c 
     ------------------------------------------------------------------------
        2      3      2                  2    2    3    2        3      2   
     - b d}, {c  - b*d , b*c - a*d, - a*c  + b d, b  - a c}, {- c  + b*d , -
     ------------------------------------------------------------------------
        2    2                   3    2        3      2       2    2    3  
     a*c  + b d, - b*c + a*d, - b  + a c}, {- c  + b*d , - a*c  + b d, b  -
     ------------------------------------------------------------------------
      2                     3      2                  2    2    3    2      4
     a c, - b*c + a*d}, {- c  + b*d , b*c - a*d, - a*c  + b d, b  - a c, - c 
     ------------------------------------------------------------------------
          3     4      3     3      2                  2    2    3    2
     + a*d }, {c  - a*d , - c  + b*d , b*c - a*d, - a*c  + b d, b  - a c}})

o7 : Sequence

i8 : weightVector(inLs#0, Ls#0)

o8 = {1, 1, 4, 6}

o8 : List

i9 : scan(#inLs, i -> print weightVector(inLs#i, Ls#i));
{1, 1, 4, 6}
{1, 1, 4, 5}
{1, 1, 3, 2}
{2, 2, 3, 1}
{4, 4, 3, 1}
{6, 6, 3, 1}
{1, 1, 2, 4}
{2, 2, 1, 2}
{6, 6, 2, 1}
{8, 8, 3, 1}

i10 : scan(#inLs, i -> print groebnerCone(inLs#i, Ls#i));
(| 0 0 |, | 1  0 |)
 | 0 0 |  | 0  1 |
 | 2 1 |  | -2 3 |
 | 3 2 |  | -3 4 |
(| 0 0 |, | 1  0 |)
 | 0 0 |  | 0  1 |
 | 2 1 |  | -2 3 |
 | 3 1 |  | -3 4 |
(| 0 0 |, | 1  0 |)
 | 0 0 |  | 0  1 |
 | 1 1 |  | -2 3 |
 | 0 1 |  | -3 4 |
(| 0 0  |, | 1  0 |)
 | 0 0  |  | 0  1 |
 | 1 0  |  | -2 3 |
 | 0 -1 |  | -3 4 |
(| 0  0  |, | 1  0 |)
 | 0  0  |  | 0  1 |
 | 0  -1 |  | -2 3 |
 | -1 -2 |  | -3 4 |
(| 0  0  |, | 1  0 |)
 | 0  0  |  | 0  1 |
 | -1 -2 |  | -2 3 |
 | -2 -3 |  | -3 4 |
(| 0 0 |, | 1  0 |)
 | 0 0 |  | 0  1 |
 | 1 0 |  | -2 3 |
 | 2 1 |  | -3 4 |
(| 0 0  |, | 1  0 |)
 | 0 0  |  | 0  1 |
 | 0 -1 |  | -2 3 |
 | 1 -1 |  | -3 4 |
(| 0  0  |, | 1  0 |)
 | 0  0  |  | 0  1 |
 | -1 -3 |  | -2 3 |
 | -1 -4 |  | -3 4 |
(| 0  0  |, | 1  0 |)
 | 0  0  |  | 0  1 |
 | -2 -3 |  | -2 3 |
 | -3 -4 |  | -3 4 |