A logarithmic form of type (d_0,...,d_n) is a differential 1-form w that can be written as w=(prod f_i)sum df_i/f_i, where f_i is a polynomial of degree d_i. This routine creates such a logarithmic form using homogeneous polynomials. When using a list L of length two, the differential form is called rational.
In this example we generate a random logarithmic form in affine 3-dimensional space with degrees (1,1,2).
i1 : random logarithmicForm(2,{1,1,2},"a")
2 2 3 2 2 2
o1 = (- 30x x + 20x x + 10x + 600x x - 395x x x - 220x x - 100x x +
0 1 0 1 1 0 2 0 1 2 1 2 0 2
------------------------------------------------------------------------
2 3 3 2 2 2
405x x - 100x )dx + (30x - 20x x - 10x x - 95x x + 80x x x +
1 2 2 0 0 0 1 0 1 0 2 0 1 2
------------------------------------------------------------------------
2 2 2 3 3 2 2
30x x + 10x x - 60x x + 15x )dx + (- 600x + 490x x + 140x x -
1 2 0 2 1 2 2 1 0 0 1 0 1
------------------------------------------------------------------------
3 2 2 2 2
30x + 100x x - 415x x x + 60x x + 100x x - 15x x )dx
1 0 2 0 1 2 1 2 0 2 1 2 2
o1 : DiffAlgForm
|
In this example we generate a generic rational form in the projective plane of type (1,1).
i2 : logarithmicForm(2,{1,1},"a",Projective => true)
o2 = ((a0 a2 a1 - a1 a0 a2 )x + (a0 a2 a1 - a1 a0 a2 )x )dx + ((-
1 0 1 0 1 1 1 1 0 2 0 1 2 2 0
------------------------------------------------------------------------
a0 a2 a1 + a1 a0 a2 )x + (a0 a1 a2 - a0 a1 a2 )x )dx + ((- a0 a2 a1
1 0 1 0 1 1 0 1 2 1 1 1 2 2 1 1 0 2
------------------------------------------------------------------------
+ a1 a0 a2 )x + (- a0 a1 a2 + a0 a1 a2 )x )dx
0 1 2 0 1 2 1 1 1 2 1 2
o2 : DiffAlgForm
|
In the following example, we produce a logarithmic form that descend to projective space.
i3 : l = random logarithmicForm(2,{1,1},"a",Projective => true)
o3 = (9x - 11x )dx + (- 9x - 19x )dx + (11x + 19x )dx
1 2 0 0 2 1 0 1 2
o3 : DiffAlgForm
|
i4 : (radial 2)_l o4 = 0 o4 : DiffAlgForm |