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Question: polynomials not in the correct indeterminates

the goal is to check kernel belong to image in Maple

restart;
with(Groebner):
K := {r-x^4,u-(x^3)*y,v-x*y^3,w-y^4};
G := Basis(K, 'tord', degrevlex(r,u,v,w));
R1 := eliminate(G, {r,u,v,w}); # eliminate is the reverse of Basis
Ga := Basis({a*G[1],a*G[2],a*G[3],a*G[4],a*G[5],a*G[6],a*G[7],a*G[8],a*G[9],a*G[10],a*G[11],a*G[12],a*G[13],a*G[14], (1-a)*K[1], (1-a)*K[2], (1-a)*K[3], (1-a)*K[4]}, 'tord', deglex(a,r,u,v,w));
Ga := remove(has, Ga, [x,y,a]);
K0 := eliminate(Ga, {r,u,v,w});

# Kernel
# r = (u*w^2)^(1/3)*u/w, u = u, v = u*w^2/(u*w^2)^(2/3), w = w
# r = (1/2)*(u*w^2)^(1/3)*(-1+I*sqrt(3))*u/w, u = u, v = 4*u*w^2/((u*w^2)^(2/3)*(-1+I*sqrt(3))^2), w = w
# r = -(1/2)*(u*w^2)^(1/3)*(1+I*sqrt(3))*u/w, u = u, v = 4*u*w^2/((u*w^2)^(2/3)*(1+I*sqrt(3))^2), w = w

K := {r-x^4,u-(x^3)*y,v-x*y^3,w-y^4};
G := Basis(K, plex(x,y,r,u,v,w));
R := Reduce(u*w^2/(u*w^2)^(2/3), G, plex(x,y,r,u,v,w), 's');
R2a := remove(has, R, [x,y]);

# Error, (in Groebner:-Reduce) polynomials not in the correct indeterminates

R := Reduce(u*w^2/(u*w^2)^(2/3), G, plex(x,y,r,u,v,w), 's');
R2b := remove(has, R, [x,y]);
 
Correct example doing is below, however above trial is failed, i am not sure whether eliminate can get kernel from basis of kernel as above example left hand side and right hand side also contain r,u,v,w
 
K := {u-x^4-x,v-x^3};
G := Basis(K, plex(x,u,v));
R := Reduce(x^5, G, plex(x,u,v), 's'); #u*x-x^2
R2 := remove(has, R, [x]);
Please Wait...