jets(ZZ,RingMap) -- the jets of a homomorphism of rings

Synopsis

Function: jets
Usage:

jets (n,f)
Inputs:
- n, an integer,
- f, a ring map,
Optional inputs:
- Projective => ..., default value false, Option for jets
Outputs:
- a ring map, obtained by applying the n-th jets functor to f

Description

This function is provided by the package Jets.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : S = QQ[t]

o2 = S

o2 : PolynomialRing

i3 : f = map(S,R,{t,t^2,t^3})

                     2   3
o3 = map (S, R, {t, t , t })

o3 : RingMap S <-- R

i4 : Jf = jets(2,f);

o4 : RingMap QQ[t0][t1][t2] <-- QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]

i5 : matrix Jf

o5 = | t2 2t0t2+t1^2 3t0^2t2+3t0t1^2 t1 2t0t1 3t0^2t1 t0 t0^2 t0^3 |

                            1                     9
o5 : Matrix (QQ[t0][t1][t2])  <-- (QQ[t0][t1][t2])

This function can also be applied when the source and/or the target of the ring homomorphism are quotients of a polynomial ring

i6 : I = ideal(y-x^2,z-x^3)

               2         3
o6 = ideal (- x  + y, - x  + z)

o6 : Ideal of R

i7 : Q = R/I

o7 = Q

o7 : QuotientRing

i8 : g = map(S,Q,{t,t^2,t^3})

                     2   3
o8 = map (S, Q, {t, t , t })

o8 : RingMap S <-- Q

i9 : isWellDefined g

o9 = true

i10 : Jg = jets(2,g);

                                                                QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
o10 : RingMap QQ[t0][t1][t2] <-- ----------------------------------------------------------------------------------------------------
                                                    2                     2            2                2       2             3
                                 (- 2x0*x2 + y2 - x1 , - 2x0*x1 + y1, - x0  + y0, - 3x0 x2 + z2 - 3x0*x1 , - 3x0 x1 + z1, - x0  + z0)

i11 : isWellDefined Jg

o11 = true

Ways to use this method: