If the argument for janetBasis is a matrix or an ideal or a Groebner basis, then J is a Janet basis for (the module generated by) M.
If the arguments for janetBasis are a chain complex and an integer, where C is the result of either janetResolution or resolution called with the optional argument 'Strategy => Involutive', then J is the Janet basis extracted from the n-th differential of C.
i1 : R = QQ[x,y]; |
i2 : I = ideal(x^3,y^2); o2 : Ideal of R |
i3 : J = janetBasis I; |
i4 : basisElements J
o4 = | y2 xy2 x3 x2y2 |
1 4
o4 : Matrix R <--- R
|
i5 : multVar J
o5 = {set {y}, set {y}, set {x, y}, set {y}}
o5 : List
|
i6 : R = QQ[x,y]; |
i7 : M = matrix {{x*y-y^3, x*y^2, x*y-x}, {x, y^2, x}};
2 3
o7 : Matrix R <--- R
|
i8 : J = janetBasis M; |
i9 : basisElements J
o9 = | y3-x xy-x x2y-x2 x3 -x x2 -x2 0 |
| 0 x x2 x2 xy-y2+x y3 x2y-xy2+x2 x3+2x2+y2 |
2 8
o9 : Matrix R <--- R
|
i10 : multVar J
o10 = {set {y}, set {y}, set {y}, set {x, y}, set {y}, set {y}, set {y}, set
-----------------------------------------------------------------------
{x, y}}
o10 : List
|
i11 : R = QQ[x,y,z]; |
i12 : I = ideal(x,y,z); o12 : Ideal of R |
i13 : C = res(I, Strategy => Involutive)
1 3 3 1
o13 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o13 : ChainComplex
|
i14 : janetBasis(C, 2)
+------+---------+
o14 = || -y ||{z, y, x}|
|| x || |
|| 0 || |
+------+---------+
|| -z ||{z, y, x}|
|| 0 || |
|| x || |
+------+---------+
|| 0 ||{z, y} |
|| -z || |
|| y || |
+------+---------+
o14 : InvolutiveBasis
|