This function is provided by the package Jets.
i1 : R= QQ[x,y] o1 = R o1 : PolynomialRing |
i2 : I= ideal (x^3 + y^3 - 3*x*y)
3 3
o2 = ideal(x + y - 3x*y)
o2 : Ideal of R
|
i3 : J= jets(3,I); o3 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3] |
i4 : netList J_*
+-----------------------------------------------------------------------------------+
| 2 2 3 3|
o4 = |(3x0 - 3y0)x3 + (3y0 - 3x0)y3 + (6x0*x1 - 3y1)x2 + (- 3x1 + 6y0*y1)y2 + x1 + y1 |
+-----------------------------------------------------------------------------------+
| 2 2 2 2 |
|(3x0 - 3y0)x2 + (3y0 - 3x0)y2 + 3x0*x1 - 3x1*y1 + 3y0*y1 |
+-----------------------------------------------------------------------------------+
| 2 2 |
|(3x0 - 3y0)x1 + (3y0 - 3x0)y1 |
+-----------------------------------------------------------------------------------+
| 3 3 |
|x0 + y0 - 3x0*y0 |
+-----------------------------------------------------------------------------------+
|
When the jets(...,Projective=>...) option is set to true, the degree of each jets variable matches its order, in accordance with Proposition 6.6 (c) of P. Vojta, Jets via Hasse-Schmidt Derivations. As a result, the jets of any ideal will be homogeneous regardless of the homogeneity of the base ideal, or that of its affine jets.
i5 : R= QQ[x,y,z] o5 = R o5 : PolynomialRing |
i6 : I= ideal (y-x^2, z-x^3)
2 3
o6 = ideal (- x + y, - x + z)
o6 : Ideal of R
|
i7 : JI= jets(2,I)
2 2 2
o7 = ideal (- 2x0*x2 + y2 - x1 , - 2x0*x1 + y1, - x0 + y0, - 3x0 x2 + z2 -
------------------------------------------------------------------------
2 2 3
3x0*x1 , - 3x0 x1 + z1, - x0 + z0)
o7 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
|
i8 : isHomogeneous JI o8 = false |
i9 : JIproj= jets(2,I,Projective=>true)
2 2 2
o9 = ideal (- 2x0*x2 + y2 - x1 , - 2x0*x1 + y1, - x0 + y0, - 3x0 x2 + z2 -
------------------------------------------------------------------------
2 2 3
3x0*x1 , - 3x0 x1 + z1, - x0 + z0)
o9 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
|
i10 : isHomogeneous JIproj o10 = true |
With Projective=>true the jet variables of order zero have degree 0, therefore no heft vector exist for the ambient ring of the jets. As a result, certain computations will not be supported, and others may run more slowly. See heft vectors for more information.