Non minimal order cross section
Counter Example to [Fels Olver Theorem 13.3 (1999)]
| fr > | Section:=[x=0,u[0]=0,v[0]=0,v[1]=0,v[2]=0, v[3]=1]:
|
| fr > | C , Invariantizations := dmf (LieAlg,Section, fr, [op(Section), u[1]=I1,v[4]=J4]): |
"Section :"
![[v[3] = 1, v[2] = 0, v[1] = 0, v[0] = 0, u[0] = 0, x = 0]](images/dmf_12.gif){kind=small}
"Transversality condition :" u[1]*u[2]
"Invariantizations:"
![x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1, v[4] = J4, u[2] = y1, u[3] = y2, u[4] = y3](images/dmf_13.gif){kind=small}
![x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1, v[4] = J4, u[2] = y1, u[3] = y2, u[4] = y3](images/dmf_14.gif){kind=small}
"Commutation rules"
"Syzygies"
![[-3*y0[0]^3*I1[0]^6-3*y0[0]^3*I1[0]^4-4*y0[0]^3*I1[0]^4*y2[0]^2+3*y0[0]^2*I1[0]^3*y2[1]+4*y0[0]^2*I1[0]^3*J4[0]*y2[0]-3*y0[0]^2*I1[0]^3*y3[0]-3*y0[0]*I1[0]*y2[0]+3*J4[0], 7*y0[0]^2*I1[0]^3*y2[0]-4*y0[...](images/dmf_15.gif){kind=small}
![[-3*y0[0]^3*I1[0]^6-3*y0[0]^3*I1[0]^4-4*y0[0]^3*I1[0]^4*y2[0]^2+3*y0[0]^2*I1[0]^3*y2[1]+4*y0[0]^2*I1[0]^3*J4[0]*y2[0]-3*y0[0]^2*I1[0]^3*y3[0]-3*y0[0]*I1[0]*y2[0]+3*J4[0], 7*y0[0]^2*I1[0]^3*y2[0]-4*y0[...](images/dmf_16.gif){kind=small}
![[-3*y0[0]^3*I1[0]^6-3*y0[0]^3*I1[0]^4-4*y0[0]^3*I1[0]^4*y2[0]^2+3*y0[0]^2*I1[0]^3*y2[1]+4*y0[0]^2*I1[0]^3*J4[0]*y2[0]-3*y0[0]^2*I1[0]^3*y3[0]-3*y0[0]*I1[0]*y2[0]+3*J4[0], 7*y0[0]^2*I1[0]^3*y2[0]-4*y0[...](images/dmf_17.gif){kind=small}
![[-3*y0[0]^3*I1[0]^6-3*y0[0]^3*I1[0]^4-4*y0[0]^3*I1[0]^4*y2[0]^2+3*y0[0]^2*I1[0]^3*y2[1]+4*y0[0]^2*I1[0]^3*J4[0]*y2[0]-3*y0[0]^2*I1[0]^3*y3[0]-3*y0[0]*I1[0]*y2[0]+3*J4[0], 7*y0[0]^2*I1[0]^3*y2[0]-4*y0[...](images/dmf_18.gif){kind=small}
![[-3*y0[0]^3*I1[0]^6-3*y0[0]^3*I1[0]^4-4*y0[0]^3*I1[0]^4*y2[0]^2+3*y0[0]^2*I1[0]^3*y2[1]+4*y0[0]^2*I1[0]^3*J4[0]*y2[0]-3*y0[0]^2*I1[0]^3*y3[0]-3*y0[0]*I1[0]*y2[0]+3*J4[0], 7*y0[0]^2*I1[0]^3*y2[0]-4*y0[...](images/dmf_19.gif){kind=small}
"Syzygies with attempt to eliminate the y's"
![y0[0] = 1/(I1[0]*y1[0])](images/dmf_20.gif){kind=small}
![y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1)](images/dmf_21.gif)
![y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*y1[0]^2*I1[1]^2+5*I1[0]*y1[0]^2*J4[0]*I1[1]+y1[0]^2*J4[0]^2+I1[0]^4*y1[0]^2*J4[0]^2+2*y1[0]^2*J4[0]^2*I1...](images/dmf_22.gif){kind=small}
![y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*y1[0]^2*I1[1]^2+5*I1[0]*y1[0]^2*J4[0]*I1[1]+y1[0]^2*J4[0]^2+I1[0]^4*y1[0]^2*J4[0]^2+2*y1[0]^2*J4[0]^2*I1...](images/dmf_23.gif){kind=small}
![y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*y1[0]^2*I1[1]^2+5*I1[0]*y1[0]^2*J4[0]*I1[1]+y1[0]^2*J4[0]^2+I1[0]^4*y1[0]^2*J4[0]^2+2*y1[0]^2*J4[0]^2*I1...](images/dmf_24.gif){kind=small}
![y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*y1[0]^2*I1[1]^2+5*I1[0]*y1[0]^2*J4[0]*I1[1]+y1[0]^2*J4[0]^2+I1[0]^4*y1[0]^2*J4[0]^2+2*y1[0]^2*J4[0]^2*I1...](images/dmf_25.gif){kind=small}
![y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*y1[0]^2*I1[1]^2+5*I1[0]*y1[0]^2*J4[0]*I1[1]+y1[0]^2*J4[0]^2+I1[0]^4*y1[0]^2*J4[0]^2+2*y1[0]^2*J4[0]^2*I1...](images/dmf_26.gif){kind=small}
![y2[0] = y1[0]*(3*I1[1]*I1[0]+J4[0]+J4[0]*I1[0]^2-3*I1[0]*y1[0])/(I1[0]^2+1)](images/dmf_27.gif)
| fr > | Invariantizations; rewrite_rules(C); |
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_28.gif)
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_29.gif){kind=small}
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_30.gif){kind=small}
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_31.gif){kind=small}
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_32.gif){kind=small}
![[x = 0, u[0] = 0, v[0] = 0, v[1] = 0, v[2] = 0, v[3] = 1, u[1] = I1[0], v[4] = J4[0], u[2] = y1[0], u[3] = y2[0], u[4] = y3[0], u[5] = 1/3*(3*y3[1]*I1[0]*y1[0]+10*y1[0]^2*J4[0]*y2[0]+5*y1[0]*J4[0]*I1[...](images/dmf_33.gif)
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_34.gif)
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_35.gif){kind=small}
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_36.gif){kind=small}
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_37.gif){kind=small}
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_38.gif){kind=small}
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_39.gif){kind=small}
![[[y0[0] = 1/(I1[0]*y1[0]), y1[1] = y1[0]*(6*I1[1]*I1[0]-6*I1[0]*y1[0]+J4[0]*I1[0]^2+J4[0])/(I1[0]^2+1), y3[0] = (-1+2*I1[0]^2*y1[0]^2*J4[1]-6*y1[0]^3*I1[1]-I1[0]^6+y1[0]^2*J4[1]+3*y1[0]^4-3*I1[0]^4+3*...](images/dmf_40.gif)