This method computes the join of two given ideals. The join can be used to describe very interesting types of primary ideals that include the symbolic powers of prime ideals. For more details reader is referred to Section 7 of the paper Primary ideals and their differential equations.
i1 : R = QQ[x_1..x_9] o1 = R o1 : PolynomialRing |
i2 : MM = genericMatrix(R, 3, 3)
o2 = | x_1 x_4 x_7 |
| x_2 x_5 x_8 |
| x_3 x_6 x_9 |
3 3
o2 : Matrix R <--- R
|
i3 : P = minors(2, MM)
o3 = ideal (- x x + x x , - x x + x x , - x x + x x , - x x + x x , -
2 4 1 5 3 4 1 6 3 5 2 6 2 7 1 8
------------------------------------------------------------------------
x x + x x , - x x + x x , - x x + x x , - x x + x x , - x x + x x )
3 7 1 9 3 8 2 9 5 7 4 8 6 7 4 9 6 8 5 9
o3 : Ideal of R
|
i4 : M = ideal(x_1^2, x_2^2, x_3^2, x_4, x_5, x_6, x_7, x_8, x_9)
2 2 2
o4 = ideal (x , x , x , x , x , x , x , x , x )
1 2 3 4 5 6 7 8 9
o4 : Ideal of R
|
i5 : Q = joinIdeals(P, M)
3 3 2 2
o5 = ideal (x x - x x , x x - x x , x x - x x , x x - 3x x x x +
6 8 5 9 6 7 4 9 5 7 4 8 3 8 2 3 8 9
------------------------------------------------------------------------
2 2 3 3 3 2 2 2 2 3 2 2 2
3x x x x - x x , x x x - 3x x x x x + 3x x x x - x x x , x x x x -
2 3 8 9 2 9 3 5 8 2 3 5 8 9 2 3 5 9 2 6 9 2 3 7 8
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2 2
x x x x - x x x x + x x x x + x x x x - x x x x , x x x x x -
1 3 7 8 2 3 7 9 1 3 8 9 1 2 7 9 1 2 8 9 2 3 4 7 8
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 3 2
x x x x - x x x x x + x x x x x + x x x x - x x x x , x x x -
1 3 4 8 2 3 4 7 9 1 3 5 8 9 1 2 4 9 1 2 5 9 3 5 8
------------------------------------------------------------------------
2 2 2 3 2 2 2 2 2 2
3x x x x + 3x x x x x - x x x , x x x x - x x x x x - x x x x +
2 3 5 9 2 3 5 6 9 2 6 9 2 3 4 8 1 3 4 5 8 2 3 4 9
------------------------------------------------------------------------
2 2 2 2 3 3 2 2 2 2 3 3
x x x x + x x x x x - x x x x x , x x - 3x x x x + 3x x x x - x x ,
1 3 5 9 1 2 4 6 9 1 2 5 6 9 3 7 1 3 7 9 1 3 7 9 1 9
------------------------------------------------------------------------
3 3 2 2 2 2 3 3 3 2 2 2 2
x x - 3x x x x + 3x x x x - x x , x x x - 3x x x x x + 3x x x x -
2 7 1 2 7 8 1 2 7 8 1 8 3 4 7 1 3 4 7 9 1 3 4 9
------------------------------------------------------------------------
3 2 3 2 2 2 2 3 2 3 2 2 2
x x x , x x x - 3x x x x x + 3x x x x - x x x , x x x - 3x x x x +
1 6 9 2 4 7 1 2 4 7 8 1 2 4 8 1 5 8 3 4 7 1 3 4 9
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 3 2 3 3
3x x x x x - x x x , x x x - 3x x x x + 3x x x x x - x x x , x x -
1 3 4 6 9 1 6 9 2 4 7 1 2 4 8 1 2 4 5 8 1 5 8 3 5
------------------------------------------------------------------------
2 2 2 2 3 3 2 2 2 2 2 2 2 2
3x x x x + 3x x x x - x x , x x x x - x x x x - x x x x + x x x x
2 3 5 6 2 3 5 6 2 6 2 3 4 5 1 3 4 5 2 3 4 6 1 3 5 6
------------------------------------------------------------------------
2 2 2 2 3 3 2 2 2 2 3 3 3 3
+ x x x x - x x x x , x x - 3x x x x + 3x x x x - x x , x x -
1 2 4 6 1 2 5 6 3 4 1 3 4 6 1 3 4 6 1 6 2 4
------------------------------------------------------------------------
2 2 2 2 3 3 2 3 2 2 4 3 2
3x x x x + 3x x x x - x x , x x x x x - x x x + x x x x x -
1 2 4 5 1 2 4 5 1 5 1 2 3 7 8 1 3 8 2 3 7 8 9
------------------------------------------------------------------------
2 2 2 3 4 2 2 3 2 2 2 2 2
4x x x x x x + 3x x x x x - x x x + 3x x x x x - 2x x x x ,
1 2 3 7 8 9 1 2 3 8 9 2 7 9 1 2 7 8 9 1 2 8 9
------------------------------------------------------------------------
2 3 2 2 3 3 2 2 2 2
x x x x x - x x x x + x x x x x x - 4x x x x x x + 3x x x x x x -
1 2 3 4 8 1 3 5 8 2 3 4 7 8 9 1 2 3 4 8 9 1 2 3 5 8 9
------------------------------------------------------------------------
4 2 3 2 2 2 2 4 2 2 3 2 2 2 2 2
x x x x + 3x x x x x - 2x x x x x , x x x - 4x x x x x - x x x x +
2 4 7 9 1 2 4 8 9 1 2 5 8 9 3 7 8 1 3 7 8 9 2 3 7 9
------------------------------------------------------------------------
2 2 2 2 2 2 2 3 2 2 4 3 2 2 2 2 3
4x x x x x x + 3x x x x - 4x x x x x + x x x , x x x x + x x x x -
1 2 3 7 8 9 1 3 8 9 1 2 3 8 9 1 2 9 1 3 7 8 2 3 7 9
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 3 2 2 2 2 3
4x x x x x - 4x x x x x + 4x x x x x x + 3x x x x + 3x x x x -
1 3 7 8 9 1 2 3 7 9 1 2 3 7 8 9 1 3 8 9 1 2 7 9
------------------------------------------------------------------------
3 3 4 2 3 2 2 2 2 2 2
4x x x x , x x x x - 4x x x x x - x x x x x + 4x x x x x x +
1 2 8 9 3 4 7 8 1 3 4 8 9 2 3 4 7 9 1 2 3 4 8 9
------------------------------------------------------------------------
2 2 2 2 3 2 2 3 3 2 2 2 2
3x x x x x - 4x x x x x + x x x x , x x x x x + x x x x x -
1 3 5 8 9 1 2 3 5 9 1 2 6 9 1 3 4 7 8 2 3 4 7 9
------------------------------------------------------------------------
2 2 2 2 2 2 2 3 2 2 2 3
4x x x x x - 4x x x x x x + 4x x x x x x + 3x x x x x + 3x x x x -
1 3 4 8 9 1 2 3 4 7 9 1 2 3 4 8 9 1 3 5 8 9 1 2 4 9
------------------------------------------------------------------------
3 3 2 2 2 2 2 2 3 2 2
4x x x x , x x x x x x - x x x x + x x x x x - 4x x x x x x x +
1 2 5 9 1 2 3 4 5 8 1 3 5 8 2 3 4 8 9 1 2 3 4 5 8 9
------------------------------------------------------------------------
2 2 4 2 2 3 2 2 2 2 2 4 2 2 3
3x x x x x x - x x x + 3x x x x x - 2x x x x , x x x - 4x x x x x x
1 2 3 5 8 9 2 4 9 1 2 4 5 9 1 2 5 9 3 4 8 1 3 4 5 8 9
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
- x x x x + 4x x x x x x + 3x x x x - 4x x x x x x + x x x x ,
2 3 4 9 1 2 3 4 5 9 1 3 5 9 1 2 3 5 6 9 1 2 6 9
------------------------------------------------------------------------
3 2 2 2 2 2 2 2 2 2 2 2 2
x x x x + x x x x x - 4x x x x x x - 4x x x x x + 4x x x x x x +
1 3 4 8 2 3 4 7 9 1 3 4 5 8 9 1 2 3 4 9 1 2 3 4 5 9
------------------------------------------------------------------------
3 2 2 2 2 2 3 2 2 2 2 2 3
3x x x x + 3x x x x x - 4x x x x x , x x x x x x - x x x x +
1 3 5 9 1 2 4 6 9 1 2 5 6 9 1 2 3 4 5 8 1 3 5 8
------------------------------------------------------------------------
3 2 2 2 2 3 4 2 3
x x x x x - 4x x x x x x + 3x x x x x - x x x x + 3x x x x x x -
2 3 4 5 9 1 2 3 4 5 9 1 2 3 5 9 2 4 6 9 1 2 4 5 6 9
------------------------------------------------------------------------
2 2 2 4 2 3 2 2 2 2 2
2x x x x x , x x x x - 4x x x x x - x x x x x + 4x x x x x x x +
1 2 5 6 9 3 4 5 8 1 3 4 5 9 2 3 4 6 9 1 2 3 4 5 6 9
------------------------------------------------------------------------
2 2 2 2 2 2 2 3 3 2 2 2 3
3x x x x x - 4x x x x x x + x x x x , x x x x x + x x x x -
1 3 5 6 9 1 2 3 5 6 9 1 2 6 9 1 3 4 5 8 2 3 4 9
------------------------------------------------------------------------
2 2 2 2 2 2 3 2
4x x x x x - 4x x x x x x + 4x x x x x x x + 3x x x x x +
1 3 4 5 9 1 2 3 4 6 9 1 2 3 4 5 6 9 1 3 5 6 9
------------------------------------------------------------------------
2 2 2 3 2 2 2 4 2 2 2 2 2 3
3x x x x x - 4x x x x x , x x x - x x x x - 4x x x x x +
1 2 4 6 9 1 2 5 6 9 2 3 7 1 3 7 8 1 2 3 7 9
------------------------------------------------------------------------
2 2 2 2 2 2 3 2 4 2 2 2 2 3 2 2 2
4x x x x x x + 3x x x x - 4x x x x x + x x x , x x x x - x x x x x
1 2 3 7 8 9 1 2 7 9 1 2 7 8 9 1 8 9 2 3 4 7 1 3 4 7 8
------------------------------------------------------------------------
2 2 2 2 2 2 3 2
- 4x x x x x x + 4x x x x x x x + 3x x x x x - 4x x x x x +
1 2 3 4 7 9 1 2 3 4 7 8 9 1 2 4 7 9 1 2 4 8 9
------------------------------------------------------------------------
4 2 2 2 2 2 2 2 2 2 2 2 2 2
x x x x , x x x x - x x x x - 4x x x x x x + 4x x x x x x +
1 5 8 9 2 3 4 7 1 3 4 8 1 2 3 4 7 9 1 2 3 4 8 9
------------------------------------------------------------------------
2 2 2 2 3 2 4 2 2 2 2 3 2 2 2 2 3
3x x x x - 4x x x x x + x x x , x x x x - x x x x x - 4x x x x x +
1 2 4 9 1 2 4 5 9 1 5 9 2 3 4 7 1 3 4 5 8 1 2 3 4 9
------------------------------------------------------------------------
2 2 2 2 2 3 4 2 2 3
4x x x x x x + 3x x x x x - 4x x x x x x + x x x x , x x x x x -
1 2 3 4 5 9 1 2 4 6 9 1 2 4 5 6 9 1 5 6 9 1 2 3 4 5
------------------------------------------------------------------------
2 2 4 3 2 2 2 2 3 4 2 2 3 2
x x x + x x x x x - 4x x x x x x + 3x x x x x - x x x + 3x x x x x
1 3 5 2 3 4 5 6 1 2 3 4 5 6 1 2 3 5 6 2 4 6 1 2 4 5 6
------------------------------------------------------------------------
2 2 2 2 4 2 2 3 2 2 2 2 2 2 2 2 2 2 2
- 2x x x x , x x x - 4x x x x x - x x x x + 4x x x x x x + 3x x x x
1 2 5 6 3 4 5 1 3 4 5 6 2 3 4 6 1 2 3 4 5 6 1 3 5 6
------------------------------------------------------------------------
2 3 2 2 4 3 2 2 2 2 3 2 2 2 2 2 2
- 4x x x x x + x x x , x x x x + x x x x - 4x x x x x - 4x x x x x
1 2 3 5 6 1 2 6 1 3 4 5 2 3 4 6 1 3 4 5 6 1 2 3 4 6
------------------------------------------------------------------------
2 2 3 2 2 2 2 3 3 3 2 2 4 2 2 2 2
+ 4x x x x x x + 3x x x x + 3x x x x - 4x x x x , x x x - x x x x -
1 2 3 4 5 6 1 3 5 6 1 2 4 6 1 2 5 6 2 3 4 1 3 4 5
------------------------------------------------------------------------
2 3 2 2 2 2 2 2 3 2 4 2 2
4x x x x x + 4x x x x x x + 3x x x x - 4x x x x x + x x x )
1 2 3 4 6 1 2 3 4 5 6 1 2 4 6 1 2 4 5 6 1 5 6
o5 : Ideal of R
|
i6 : isPrimary Q o6 = true |
The object joinIdeals is a method function.