To select terms of a single degree, use part(deg, f). An alternate syntax uses an underscore.
i1 : R = QQ[x,y]; |
i2 : f = (x+y+1)^4
4 3 2 2 3 4 3 2 2 3 2
o2 = x + 4x y + 6x y + 4x*y + y + 4x + 12x y + 12x*y + 4y + 6x +
------------------------------------------------------------------------
2
12x*y + 6y + 4x + 4y + 1
o2 : R
|
i3 : part(2, f)
2 2
o3 = 6x + 12x*y + 6y
o3 : R
|
i4 : part_2 f
2 2
o4 = 6x + 12x*y + 6y
o4 : R
|
To select terms within a range of degrees, use part(lo, hi, f).
i5 : part(1, 2, f)
2 2
o5 = 6x + 12x*y + 6y + 4x + 4y
o5 : R
|
In the next example, we apply the weights \{2,3\} before selecting terms. In other words, the term x^ay^b is considered to have degree 2a+3b.
i6 : part(6, {2,3}, f)
3 2
o6 = 4x + 6y
o6 : R
|
i7 : part(6, 8, {2,3}, f)
4 3 2 2 2
o7 = x + 4x + 12x y + 12x*y + 6y
o7 : R
|
If the generators of the ring were defined to have non-unit degrees, the weights override those degrees.
i8 : R = QQ[x,y, Degrees=>{2,3}];
|
i9 : f = (x+y+1)^4
4 3 2 2 3 3 4 2 2 3 2
o9 = y + 4x*y + 6x y + 4x y + 4y + x + 12x*y + 12x y + 4x + 6y +
------------------------------------------------------------------------
2
12x*y + 6x + 4y + 4x + 1
o9 : R
|
i10 : part(2, f) o10 = 4x o10 : R |
i11 : part(2, {1,1}, f)
2 2
o11 = 6y + 12x*y + 6x
o11 : R
|
By omitting lo or hi, but providing a comma indicating the omission, the range of degrees will be unbounded in the appropriate direction.
i12 : S = QQ[a,b,c] o12 = S o12 : PolynomialRing |
i13 : g = (a - b*c + 2)^3
3 3 2 2 2 2 2 3 2
o13 = - b c + 3a*b c - 3a b*c + 6b c + a - 12a*b*c + 6a - 12b*c + 12a +
-----------------------------------------------------------------------
8
o13 : S
|
i14 : part(4, , g)
3 3 2 2 2 2 2
o14 = - b c + 3a*b c - 3a b*c + 6b c
o14 : S
|
i15 : part(, 3, g)
3 2
o15 = a - 12a*b*c + 6a - 12b*c + 12a + 8
o15 : S
|
i16 : part(, 3, 1..3, g)
3 2
o16 = a + 6a + 12a + 8
o16 : S
|
Infinite numbers may also be given for the bounds.
i17 : part(4, infinity, g)
3 3 2 2 2 2 2
o17 = - b c + 3a*b c - 3a b*c + 6b c
o17 : S
|
i18 : part(-infinity, 3, g)
3 2
o18 = a - 12a*b*c + 6a - 12b*c + 12a + 8
o18 : S
|
i19 : part(-infinity, infinity, 1..3, g)
3 3 2 2 2 2 2 3 2
o19 = - b c + 3a*b c - 3a b*c + 6b c + a - 12a*b*c + 6a - 12b*c + 12a +
-----------------------------------------------------------------------
8
o19 : S
|
For multigraded rings, use a list to specify a single multidegree in the first argument. The underscore syntax works here too.
i20 : R = QQ[x,y,z, Degrees => {{1,0,0},{0,1,0},{0,0,1}}];
|
i21 : f = (x+y+z)^3
3 2 2 3 2 2 2 2 3
o21 = x + 3x y + 3x*y + y + 3x z + 6x*y*z + 3y z + 3x*z + 3y*z + z
o21 : R
|
i22 : part({2,0,1}, f)
2
o22 = 3x z
o22 : R
|
i23 : part_{2,0,1} f
2
o23 = 3x z
o23 : R
|
A range of degrees cannot be asked for in the multigraded case. Polynomial rings over polynomial rings are multigraded, so either use a multidegree or specify weights to avoid errors.
i24 : R = QQ[a][x]; |
i25 : h = (1+a+x)^3
3 2 2 3 2
o25 = x + (3a + 3)x + (3a + 6a + 3)x + a + 3a + 3a + 1
o25 : R
|
i26 : part(2, {1,0}, h)
2
o26 = (3a + 3)x
o26 : R
|
i27 : part(2, {0,1}, h)
2 2
o27 = 3a x + 3a
o27 : R
|
i28 : part({2,1}, h)
2
o28 = (3a + 3)x
o28 : R
|
The object part is a method function.