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GenericInitialIdeal :: gin

gin -- the generic initial ideal

Synopsis

Description

Example: a complete intersection of type (3,3) in P^3

i1 : R = QQ[a..d];
i2 : I = ideal(a^3+c^2*d, b^3-a*d^2);

o2 : Ideal of R
i3 : gin(I)

             3   2      3   5
o3 = ideal (a , a b, a*b , b )

o3 : Ideal of R

The Stanley-Reisner ideal of RP^2

i4 : R = QQ[x0,x1,x2,x3,x4,x5]

o4 = R

o4 : PolynomialRing
i5 : M = matrix {{x1*x3*x4, x0*x3*x4, x1*x2*x4, x0*x2*x3, x0*x1*x2, x2*x4*x5, x0*x4*x5, x2*x3*x5, x1*x3*x5, x0*x1*x5}} --Stanley-Reisner ideal of RP^2

o5 = | x1x3x4 x0x3x4 x1x2x4 x0x2x3 x0x1x2 x2x4x5 x0x4x5 x2x3x5 x1x3x5 x0x1x5
     ------------------------------------------------------------------------
     |

             1       10
o5 : Matrix R  <--- R
i6 : I=ideal flatten entries M

o6 = ideal (x1*x3*x4, x0*x3*x4, x1*x2*x4, x0*x2*x3, x0*x1*x2, x2*x4*x5,
     ------------------------------------------------------------------------
     x0*x4*x5, x2*x3*x5, x1*x3*x5, x0*x1*x5)

o6 : Ideal of R
i7 : J=(ideal{x0,x1,x2})^3

              3    2      2         2                 2    3    2         2 
o7 = ideal (x0 , x0 x1, x0 x2, x0*x1 , x0*x1*x2, x0*x2 , x1 , x1 x2, x1*x2 ,
     ------------------------------------------------------------------------
       3
     x2 )

o7 : Ideal of R
i8 : assert(gin(I)==J)

Example 1.10 from Conca, De Negri, Gorla 'Cartwright-Sturmfels ideals associated to graphs and linear spaces'.

i9 : R = QQ[x_1..x_3,y_1..y_3, Degrees=>{{1,0},{1,0},{1,0},{0,1},{0,1},{0,1}}];
i10 : I = ideal(x_1*y_1,x_2*y_2,x_3*y_2,x_2*y_3,x_3*y_3);

o10 : Ideal of R
i11 : gin(I)

              2         2               3   2
o11 = ideal (x , x x , x , x x , x x , x , x y )
              1   1 2   2   1 3   2 3   3   3 1

o11 : Ideal of R
i12 : gin(I, Multigraded => true)

                                            2
o12 = ideal (x y , x y , x y , x y , x y , x y , x x y )
              1 1   2 1   3 1   1 2   2 2   1 3   1 2 3

o12 : Ideal of R

This symbol is provided by the package GenericInitialIdeal.

Caveat

The method gin uses a probabilistic algorithm. The returned answer is correct with high probability in characteristic zero and large positive characteristic, but might be wrong in small positive characteristic. For details in this situation it is recommended to use the Verbose option.

See also

Ways to use gin :

For the programmer

The object gin is a method function with options.