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Jets > jets > jets(ZZ,RingMap)

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

Synopsis

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: