restrictedUnfolding -- Compute a restricted unfolding

Usage:

(A,runfolding)=restrictedUnfolding(I,positionList)
Inputs:
- I, an ideal, ideal of semigroup ring
- positionList, a list, list of positions of the unfolding parameters to be used in the restricted unfolding
Outputs:
- A, a ring, coordinate ring of the restricted unfolding
- runfolding, a matrix, equations of the restricted unfolding

Description

Given ideal c the functions adds random linear equations L to c to obtain 1-dimensional ideal. Since the ground field is finite, decompose the ideal c+L will lead to a point with positive probability. Thus repeating will lead to success.

i1 : L={5,6,8}

o1 = {5, 6, 8}

o1 : List

i2 : I=ideal semigroupRing(L,"BaseField"=>ZZ/nextPrime 10^4);

                ZZ
o2 : Ideal of -----[x ..x , x ]
              10007  0   1   3

i3 : (A,unfolding)=makeUnfolding I;

i4 : gens A/degree

o4 = {{20}, {18}, {16}, {15}, {14}, {13}, {12}, {12}, {11}, {10}, {10}, {10},
     ------------------------------------------------------------------------
     {9}, {8}, {8}, {8}, {7}, {7}, {6}, {6}, {6}, {5}, {5}, {5}, {4}, {4},
     ------------------------------------------------------------------------
     {4}, {3}, {3}, {3}, {2}, {2}, {2}, {1}, {1}, {1}}

o4 : List

i5 : numgens A

o5 = 36

i6 : positionList=toList(0..30)

o6 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
     ------------------------------------------------------------------------
     20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

o6 : List

i7 : (A,runfolding)=restrictedUnfolding(I,positionList)

o7 = (A, | x_0^2x_1-x_3^2+a_{0, 0}+x_0a_{0, 1}+x_1a_{0, 2}+x_3a_{0,
     ------------------------------------------------------------------------
     3}+x_0^2a_{0, 4}+x_0x_1a_{0, 5}+x_1^2a_{0, 6}+x_0x_3a_{0, 7}
     ------------------------------------------------------------------------
     x_1^3-x_0^2x_3+a_{1, 0}+x_0a_{1, 1}+x_1a_{1, 2}+x_3a_{1, 3}+x_0^2a_{1,
     ------------------------------------------------------------------------
     4}+x_0x_1a_{1, 5}+x_1^2a_{1, 6}+x_0x_3a_{1, 7}+x_1x_3a_{1, 8}+x_0^3a_{1,
     ------------------------------------------------------------------------
     9} x_0^4-x_1^2x_3+a_{2, 0}+x_0a_{2, 1}+x_1a_{2, 2}+x_3a_{2,
     ------------------------------------------------------------------------
     3}+x_0^2a_{2, 4}+x_0x_1a_{2, 5}+x_1^2a_{2, 6}+x_0x_3a_{2, 7}+x_1x_3a_{2,
     ------------------------------------------------------------------------
     8}+x_0^3a_{2, 9}+x_3^2a_{2, 10}+x_0x_1^2a_{2, 11}+x_0^2x_3a_{2, 12} |)

o7 : Sequence

i8 : (J,family)=getFlatFamily(I,A,runfolding)

o8 = (ideal (), | x_0^2x_1-x_3^2+x_1a_{0, 2}+a_{1, 4}^2+x_3a_{0, 3}-a_{1,
     ------------------------------------------------------------------------
     4}a_{0, 3}+x_0^2a_{0, 4}+a_{0, 2}a_{0, 4}+x_0x_1a_{0, 5}+x_0a_{0,
     ------------------------------------------------------------------------
     4}a_{0, 5}+x_1^2a_{0, 6}-a_{0, 4}^2a_{0, 6}+2x_0a_{1, 4}a_{1,
     ------------------------------------------------------------------------
     9}-x_0a_{0, 3}a_{1, 9}+2a_{1, 4}a_{1, 7}a_{1, 9}-a_{0, 3}a_{1, 7}a_{1,
     ------------------------------------------------------------------------
     9}-a_{0, 2}a_{1, 9}^2+2x_0a_{1, 7}a_{1, 9}^2+a_{1, 7}^2a_{1,
     ------------------------------------------------------------------------
     9}^2-x_0a_{0, 5}a_{1, 9}^2+2a_{0, 4}a_{0, 6}a_{1, 9}^2-a_{0, 6}a_{1,
     ------------------------------------------------------------------------
     9}^4+x_0x_3a_{0, 7}-x_0a_{1, 4}a_{0, 7}+a_{0, 2}a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}-x_0a_{1, 7}a_{1, 9}a_{0, 7}+x_0a_{0, 5}a_{1, 9}a_{0, 7}-2a_{0,
     ------------------------------------------------------------------------
     4}a_{0, 6}a_{1, 9}a_{0, 7}+2a_{0, 6}a_{1, 9}^3a_{0, 7}-a_{0, 6}a_{1,
     ------------------------------------------------------------------------
     9}^2a_{0, 7}^2 x_1^3-x_0^2x_3+x_1a_{1, 2}+x_3a_{1, 3}+x_0^2a_{1,
     ------------------------------------------------------------------------
     4}-a_{1, 3}a_{1, 4}+x_0x_1a_{1, 5}+x_1^2a_{1, 6}+a_{1, 2}a_{0,
     ------------------------------------------------------------------------
     4}+x_0a_{1, 5}a_{0, 4}-a_{1, 6}a_{0, 4}^2+a_{0, 4}^3+x_0x_3a_{1,
     ------------------------------------------------------------------------
     7}-x_0a_{1, 4}a_{1, 7}+x_1x_3a_{1, 8}+a_{1, 4}a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     8}+x_0^3a_{1, 9}-x_0a_{1, 3}a_{1, 9}-a_{1, 3}a_{1, 7}a_{1, 9}-x_0a_{1,
     ------------------------------------------------------------------------
     7}^2a_{1, 9}+x_0a_{0, 4}a_{1, 8}a_{1, 9}+a_{0, 4}a_{1, 7}a_{1, 8}a_{1,
     ------------------------------------------------------------------------
     9}-a_{1, 2}a_{1, 9}^2-x_0a_{1, 5}a_{1, 9}^2+2a_{1, 6}a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     9}^2-3a_{0, 4}^2a_{1, 9}^2-a_{1, 4}a_{1, 8}a_{1, 9}^2-x_0a_{1, 8}a_{1,
     ------------------------------------------------------------------------
     9}^3-a_{1, 7}a_{1, 8}a_{1, 9}^3-a_{1, 6}a_{1, 9}^4+3a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     9}^4-a_{1, 9}^6+a_{1, 2}a_{1, 9}a_{0, 7}+x_0a_{1, 5}a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}-2a_{1, 6}a_{0, 4}a_{1, 9}a_{0, 7}+3a_{0, 4}^2a_{1, 9}a_{0, 7}+a_{1,
     ------------------------------------------------------------------------
     4}a_{1, 8}a_{1, 9}a_{0, 7}+x_0a_{1, 8}a_{1, 9}^2a_{0, 7}+a_{1, 7}a_{1,
     ------------------------------------------------------------------------
     8}a_{1, 9}^2a_{0, 7}+2a_{1, 6}a_{1, 9}^3a_{0, 7}-6a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     9}^3a_{0, 7}+3a_{1, 9}^5a_{0, 7}-a_{1, 6}a_{1, 9}^2a_{0, 7}^2+3a_{0,
     ------------------------------------------------------------------------
     4}a_{1, 9}^2a_{0, 7}^2-3a_{1, 9}^4a_{0, 7}^2+a_{1, 9}^3a_{0, 7}^3
     ------------------------------------------------------------------------
     x_0^4-x_1^2x_3-x_3a_{1, 2}-x_0^2a_{1, 3}+x_0^2a_{0, 2}-a_{1, 3}a_{0,
     ------------------------------------------------------------------------
     2}-x_1^2a_{1, 4}-a_{1, 2}a_{1, 4}+x_1^2a_{0, 3}+a_{1, 2}a_{0,
     ------------------------------------------------------------------------
     3}-x_0x_3a_{1, 5}-x_0a_{1, 4}a_{1, 5}+x_0a_{0, 3}a_{1, 5}-x_1x_3a_{1,
     ------------------------------------------------------------------------
     6}-x_1a_{1, 4}a_{1, 6}+x_1a_{0, 3}a_{1, 6}+x_1x_3a_{0, 4}+x_1a_{1,
     ------------------------------------------------------------------------
     4}a_{0, 4}-x_1a_{0, 3}a_{0, 4}+x_3a_{1, 6}a_{0, 4}+a_{1, 4}a_{1, 6}a_{0,
     ------------------------------------------------------------------------
     4}-a_{0, 3}a_{1, 6}a_{0, 4}-x_3a_{0, 4}^2-a_{1, 4}a_{0, 4}^2+a_{0,
     ------------------------------------------------------------------------
     3}a_{0, 4}^2-x_0^3a_{1, 7}-x_0a_{0, 2}a_{1, 7}+x_0^3a_{0, 5}-x_0a_{1,
     ------------------------------------------------------------------------
     3}a_{0, 5}-x_0^2a_{1, 7}a_{0, 5}-x_3^2a_{1, 8}-x_3a_{1, 4}a_{1,
     ------------------------------------------------------------------------
     8}+x_3a_{0, 3}a_{1, 8}+x_0^2a_{0, 4}a_{1, 8}+a_{0, 2}a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     8}+x_0a_{0, 4}a_{0, 5}a_{1, 8}+x_3^2a_{0, 6}-x_1a_{1, 3}a_{0,
     ------------------------------------------------------------------------
     6}-x_1a_{0, 2}a_{0, 6}-a_{1, 4}^2a_{0, 6}-x_3a_{0, 3}a_{0, 6}+a_{1,
     ------------------------------------------------------------------------
     4}a_{0, 3}a_{0, 6}-2x_0^2a_{0, 4}a_{0, 6}+a_{1, 3}a_{0, 4}a_{0, 6}-a_{0,
     ------------------------------------------------------------------------
     2}a_{0, 4}a_{0, 6}-x_0x_1a_{1, 7}a_{0, 6}+x_0a_{0, 4}a_{1, 7}a_{0,
     ------------------------------------------------------------------------
     6}-x_0x_1a_{0, 5}a_{0, 6}-x_0a_{0, 4}a_{0, 5}a_{0, 6}+x_1a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     8}a_{0, 6}-a_{0, 4}^2a_{1, 8}a_{0, 6}-x_1^2a_{0, 6}^2+a_{0, 4}^2a_{0,
     ------------------------------------------------------------------------
     6}^2-x_0x_1^2a_{1, 9}-x_0a_{1, 2}a_{1, 9}-x_0^2a_{1, 5}a_{1,
     ------------------------------------------------------------------------
     9}-x_0x_1a_{1, 6}a_{1, 9}+x_0x_1a_{0, 4}a_{1, 9}+x_0a_{1, 6}a_{0,
     ------------------------------------------------------------------------
     4}a_{1, 9}-x_0a_{0, 4}^2a_{1, 9}-x_1^2a_{1, 7}a_{1, 9}-a_{1, 2}a_{1,
     ------------------------------------------------------------------------
     7}a_{1, 9}-x_0a_{1, 5}a_{1, 7}a_{1, 9}-x_1a_{1, 6}a_{1, 7}a_{1,
     ------------------------------------------------------------------------
     9}+x_1a_{0, 4}a_{1, 7}a_{1, 9}+a_{1, 6}a_{0, 4}a_{1, 7}a_{1, 9}-a_{0,
     ------------------------------------------------------------------------
     4}^2a_{1, 7}a_{1, 9}-x_0x_3a_{1, 8}a_{1, 9}-x_3a_{1, 7}a_{1, 8}a_{1,
     ------------------------------------------------------------------------
     9}-2x_0a_{1, 4}a_{0, 6}a_{1, 9}+x_0a_{0, 3}a_{0, 6}a_{1, 9}-2a_{1,
     ------------------------------------------------------------------------
     4}a_{1, 7}a_{0, 6}a_{1, 9}+a_{0, 3}a_{1, 7}a_{0, 6}a_{1, 9}-x_1x_3a_{1,
     ------------------------------------------------------------------------
     9}^2-x_1a_{1, 4}a_{1, 9}^2+x_1a_{0, 3}a_{1, 9}^2-x_3a_{1, 6}a_{1,
     ------------------------------------------------------------------------
     9}^2-a_{1, 4}a_{1, 6}a_{1, 9}^2+a_{0, 3}a_{1, 6}a_{1, 9}^2+2x_3a_{0,
     ------------------------------------------------------------------------
     4}a_{1, 9}^2+2a_{1, 4}a_{0, 4}a_{1, 9}^2-2a_{0, 3}a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     9}^2-x_0^2a_{1, 8}a_{1, 9}^2-a_{0, 2}a_{1, 8}a_{1, 9}^2-x_0a_{0, 5}a_{1,
     ------------------------------------------------------------------------
     8}a_{1, 9}^2+x_0^2a_{0, 6}a_{1, 9}^2-a_{1, 3}a_{0, 6}a_{1, 9}^2+a_{0,
     ------------------------------------------------------------------------
     2}a_{0, 6}a_{1, 9}^2-3x_0a_{1, 7}a_{0, 6}a_{1, 9}^2-a_{1, 7}^2a_{0,
     ------------------------------------------------------------------------
     6}a_{1, 9}^2+x_0a_{0, 5}a_{0, 6}a_{1, 9}^2-x_1a_{1, 8}a_{0, 6}a_{1,
     ------------------------------------------------------------------------
     9}^2+2a_{0, 4}a_{1, 8}a_{0, 6}a_{1, 9}^2-2a_{0, 4}a_{0, 6}^2a_{1,
     ------------------------------------------------------------------------
     9}^2-x_0x_1a_{1, 9}^3-x_0a_{1, 6}a_{1, 9}^3+2x_0a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     9}^3-x_1a_{1, 7}a_{1, 9}^3-a_{1, 6}a_{1, 7}a_{1, 9}^3+2a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     7}a_{1, 9}^3-x_3a_{1, 9}^4-a_{1, 4}a_{1, 9}^4+a_{0, 3}a_{1, 9}^4-a_{1,
     ------------------------------------------------------------------------
     8}a_{0, 6}a_{1, 9}^4+a_{0, 6}^2a_{1, 9}^4-x_0a_{1, 9}^5-a_{1, 7}a_{1,
     ------------------------------------------------------------------------
     9}^5+x_0x_1^2a_{0, 7}+x_0a_{1, 2}a_{0, 7}+x_0^2a_{1, 5}a_{0,
     ------------------------------------------------------------------------
     7}+x_0x_1a_{1, 6}a_{0, 7}-x_0x_1a_{0, 4}a_{0, 7}-x_0a_{1, 6}a_{0,
     ------------------------------------------------------------------------
     4}a_{0, 7}+x_0a_{0, 4}^2a_{0, 7}+x_0x_3a_{1, 8}a_{0, 7}-x_0x_3a_{0,
     ------------------------------------------------------------------------
     6}a_{0, 7}+x_0a_{1, 4}a_{0, 6}a_{0, 7}+x_1x_3a_{1, 9}a_{0, 7}+x_1a_{1,
     ------------------------------------------------------------------------
     4}a_{1, 9}a_{0, 7}-x_1a_{0, 3}a_{1, 9}a_{0, 7}+x_3a_{1, 6}a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}+a_{1, 4}a_{1, 6}a_{1, 9}a_{0, 7}-a_{0, 3}a_{1, 6}a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}-2x_3a_{0, 4}a_{1, 9}a_{0, 7}-2a_{1, 4}a_{0, 4}a_{1, 9}a_{0, 7}+2a_{0,
     ------------------------------------------------------------------------
     3}a_{0, 4}a_{1, 9}a_{0, 7}+x_0^2a_{1, 8}a_{1, 9}a_{0, 7}+a_{0, 2}a_{1,
     ------------------------------------------------------------------------
     8}a_{1, 9}a_{0, 7}+x_0a_{0, 5}a_{1, 8}a_{1, 9}a_{0, 7}-x_0^2a_{0,
     ------------------------------------------------------------------------
     6}a_{1, 9}a_{0, 7}+a_{1, 3}a_{0, 6}a_{1, 9}a_{0, 7}-a_{0, 2}a_{0,
     ------------------------------------------------------------------------
     6}a_{1, 9}a_{0, 7}+2x_0a_{1, 7}a_{0, 6}a_{1, 9}a_{0, 7}-x_0a_{0, 5}a_{0,
     ------------------------------------------------------------------------
     6}a_{1, 9}a_{0, 7}+x_1a_{1, 8}a_{0, 6}a_{1, 9}a_{0, 7}-2a_{0, 4}a_{1,
     ------------------------------------------------------------------------
     8}a_{0, 6}a_{1, 9}a_{0, 7}+2a_{0, 4}a_{0, 6}^2a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}+2x_0x_1a_{1, 9}^2a_{0, 7}+2x_0a_{1, 6}a_{1, 9}^2a_{0, 7}-4x_0a_{0,
     ------------------------------------------------------------------------
     4}a_{1, 9}^2a_{0, 7}+x_1a_{1, 7}a_{1, 9}^2a_{0, 7}+a_{1, 6}a_{1, 7}a_{1,
     ------------------------------------------------------------------------
     9}^2a_{0, 7}-2a_{0, 4}a_{1, 7}a_{1, 9}^2a_{0, 7}+2x_3a_{1, 9}^3a_{0,
     ------------------------------------------------------------------------
     7}+2a_{1, 4}a_{1, 9}^3a_{0, 7}-2a_{0, 3}a_{1, 9}^3a_{0, 7}+2a_{1,
     ------------------------------------------------------------------------
     8}a_{0, 6}a_{1, 9}^3a_{0, 7}-2a_{0, 6}^2a_{1, 9}^3a_{0, 7}+3x_0a_{1,
     ------------------------------------------------------------------------
     9}^4a_{0, 7}+2a_{1, 7}a_{1, 9}^4a_{0, 7}-x_0x_1a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}^2-x_0a_{1, 6}a_{1, 9}a_{0, 7}^2+2x_0a_{0, 4}a_{1, 9}a_{0,
     ------------------------------------------------------------------------
     7}^2-x_3a_{1, 9}^2a_{0, 7}^2-a_{1, 4}a_{1, 9}^2a_{0, 7}^2+a_{0, 3}a_{1,
     ------------------------------------------------------------------------
     9}^2a_{0, 7}^2-a_{1, 8}a_{0, 6}a_{1, 9}^2a_{0, 7}^2+a_{0, 6}^2a_{1,
     ------------------------------------------------------------------------
     9}^2a_{0, 7}^2-3x_0a_{1, 9}^3a_{0, 7}^2-a_{1, 7}a_{1, 9}^3a_{0,
     ------------------------------------------------------------------------
     7}^2+x_0a_{1, 9}^2a_{0, 7}^3 |)

o8 : Sequence

i9 : support runfolding

o9 = {x , x , x , a      , a      , a      , a      , a      , a      , a   
       0   1   3   {2, 0}   {1, 0}   {0, 0}   {2, 1}   {2, 2}   {1, 1}   {2,
     ------------------------------------------------------------------------
       , a      , a      , a      , a      , a      , a      , a      , a   
     3}   {1, 2}   {0, 1}   {2, 4}   {1, 3}   {0, 2}   {2, 5}   {2, 6}   {1,
     ------------------------------------------------------------------------
       , a      , a      , a      , a      , a      , a      , a      , a   
     4}   {0, 3}   {2, 7}   {1, 5}   {2, 8}   {1, 6}   {0, 4}   {2, 9}   {1,
     ------------------------------------------------------------------------
       , a      , a       , a      , a      , a       , a      , a      ,
     7}   {0, 5}   {2, 10}   {1, 8}   {0, 6}   {2, 11}   {1, 9}   {0, 7} 
     ------------------------------------------------------------------------
     a       }
      {2, 12}

o9 : List

i10 : support family

o10 = {x , x , x , a      , a      , a      , a      , a      , a      , a   
        0   1   3   {1, 2}   {1, 3}   {0, 2}   {1, 4}   {0, 3}   {1, 5}   {1,
      -----------------------------------------------------------------------
        , a      , a      , a      , a      , a      , a      , a      }
      6}   {0, 4}   {1, 7}   {0, 5}   {1, 8}   {0, 6}   {1, 9}   {0, 7}

o10 : List

Ways to use restrictedUnfolding:

restrictedUnfolding(Ideal,List)

For the programmer

The object restrictedUnfolding is a method function.

The source of this document is in WeierstrassSemigroups.m2:1467:0.