ComputerUser

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when calculate the intersection of I2 and J2, i exactly follow this picture
https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!311

however, it is G list which is not the stated in this picture

restart;
with(Groebner):
I2 := {x^2+y^3-1, x-y*x+3};
J2 := {y*x^2-1};
G1 := Basis({expand(w*I2[1]),expand(w*I2[2]), expand((1-w)*J2[1])}, 'tord', deglex(x,y));
G2 := Basis({expand(w*I2[1]),expand(w*I2[2]), expand((1-w)*J2[1])}, 'tord', deglex(w,x,y));

the goal is write a for loop to iterate and get the power of a variable with degree function

i use nops and convert(F, list) can access each terms and variables

however y^2 can be convert to list [y,2] 

hope to add if condition to stop convert this.

how to do better?

i just guess...

r-x^4
u-(x^3)*y
v-x*y^3
w-y^4

Step 1
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));

Step 2
G intersect k[r,u,v,w]

G2 := Basis({w*G[1],w*G[2],w*G[3],w*G[4],w*G[5],w*G[6],w*G[7],w*G[8],w*G[9],w*G[10],w*G[11],w*G[12],w*G[13],w*G[14], (1-w)*K[1], (1-w)*K[2], (1-w)*K[3], (1-w)*K[4]}, 'tord', degrevlex(r,u,v,w));

G2 has too much though include the correct solution

correct solution...
g1 := x^2-w*y;
g2 := x*y-w*z;
g3 := y^2-x*z;
S12 := SPolynomial(g1, g2, plex(x, y, z, w));
eq1:= S12 = u1*g1 + u2*g2 + u3*g3;
with(LinearAlgebra):
T:=GenerateMatrix([eq1],[u1,u2, u3],augmented);
S:=LinearSolve(T,free=t);

(u1,u2,u3) should be (0,0,-w);
 
the goal is find a u1,u2,u3 to make eq1 equal S12, where u1,u2,u3 should be in terms of x y z w
g3 := y^2-x*z;
X3 := LeadingPowerOfProduct(g3, plex(x,y,z,w));

X3 should be y^2
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