restart;
with(Groebner):
g1 := x^2-w*y;
g2 := x*y-w*z;
g3 := y^2-x*z;
S13 := -y^3*w+x^3*z;
eq1:= S13 = h131*g1 + h132*g2 + h133*g3;

hsol := solve(identity(eq1, x), [h131, h132, h133]);
match(eq1,{x,y,z,w},'s');
s;

h131 should be x*z

h312 should be 0

h133 should be -y*w

`if`(p::`+`, nops, 1)(p);

'if'(p::'w', nops,1)(p);

w not exist

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...