# jets(ZZ,Ideal) -- compute jets of a an ideal in a polynomial ring

## Description

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

## Caveat

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.