LIB "graal.lib";
ring Q = 0,(x,y,z,w),dp;
ideal circle = (x-1)^2+y^2-3,z;
ideal twistedCubic = xz-y2,yw-z2,xw-yz,z;
ideal I = std(intersect(circle,twistedCubic));
// the resolution is more complicated due to the twisted cubic
res(I,0);
==>  1      4      5      2      
==> Q <--  Q <--  Q <--  Q
==> 
==> 0      1      2      3      
==> resolution not minimized yet
==> 
// however if we localize outside of the twisted cubic,
// it should become very easy again.
ideal L = std(I+ideal(x-1));
graalBearer Gr = graalMixed(L); Gr;
==> affine coordinate ring: 
==>    (0),(x,y,z,w),(dp(4),C)
==> 
==> ideal defining the subvariety: 
==>    <z,x-1,y2w-3w,y4-3y2>
==> 
==> Al: 
==>    (0,w),(Y(1),Y(2),Y(3),Y(4),x,y,z),(ds(4),c,dp(3))
==>      mod <(w)*y^2+(-3*w)-Y(3),x-1-Y(2),z-Y(1),(3*w)*Y(3)+(-w^2)*Y(4)+Y(3)\
   ^2>
==> graal: 
==>    (0,w),(Y(1),Y(2),Y(3),Y(4),z),(c,dp(4),lp(1))
==>      mod <3*Y(3)+(-w)*Y(4),z^2+10*z-2>
==>    where 
==>      Y(1) represents generator z
==>      Y(2) represents generator x-1
==>      Y(3) represents generator y2w-3w
==>      Y(4) represents generator y4-3y2
==>    and x,y,z in Al are mapped to 1,1/3*z+5/3,0 in Graal
==> 
markedResolution mr = resolutionInLocalization(I,Gr);
==> // ** full resolution in a qring may be infinite, setting max length to 5
mr;
==> resolution over Al:
==>   1       2       1       
==> Al <--  Al <--  Al
==> 
==> 0       1       2       
==> resolution not minimized yet
==> 
==> k=1
==> Y(1),(w^2)*Y(4)+(3*w^2)*Y(2)^2*x+(3*w)*Y(2)*Y(3)-Y(3)^2
==> 
==> k=2
==> _[1,1],
==> -Y(1)  
==> 
==> resolution over Graal:
==>      1          2          1          
==> Graal <--  Graal <--  Graal
==> 
==> 0          1          2          
==> 
==> k=1
==> Y(1),(w^2)*Y(4)
==> 
==> k=2
==> (w^2)*Y(4),
==> -Y(1)      
==> 
==>