k=QQ
R=k[X,x_1..x_3,E1,E2,E3,MonomialOrder=>Lex]
e1=x_1+x_2+x_3
e2=x_1*x_2+x_1*x_3+x_2*x_3
e3=x_1*x_2*x_3

w=(x_1-x_2)^2*(x_2-x_3)^2*(x_1-x_3)^2

i1=x_1^4*x_2*x_3+x_1*x_2^4*x_3+x_1*x_2*x_3^4
i2=x_1^4*x_2^2+x_1^4*x_3^2+x_1^2*x_2^4+x_1^2*x_3^4+x_2^4*x_3^2+x_2^2*x_3^4
i3=x_1^3*x_2^3+x_1^3*x_3^3+x_2^3*x_3^3
i4=x_1^3*x_2^2*x_3+x_1^3*x_2*x_3^2+x_1^2*x_2^3*x_3+x_1^2*x_2*x_3^3+x_1*x_2^3*x_3^2+x_1*x_2^2*x_3^3
i5=x_1^2*x_2^2*x_3^2

S=k[W,I1,I2,I3,I4,I5,Degrees=>{6,6,6,6,6,6}]
M=matrix({{w,i1,i2,i3,i4,i5}});
Phi=map(R,S,M)
ker Phi
transpose mingens ker Phi
-- also gilt:
w==-(2*i1-i2+2*i3-2*i4+6*i5)


S=k[E1,E2,E3,I,Degrees=>{1,2,3,6}]
M=matrix({{e1,e2,e3,i1}});
Phi=map(R,S,M);
ker Phi
-- also gilt
i1==e1^3*e3-3*e1*e2*e3+3*e3^2


M=matrix({{e1,e2,e3,i2}});
Phi=map(R,S,M);
ker Phi
-- also gilt
i2==e1^2*e2^2-2*e2^3-2*e1^3*e3+4*e1*e2*e3-3*e3^2

M=matrix({{e1,e2,e3,i3}});
Phi=map(R,S,M);
ker Phi
-- also gilt
i3==e2^3-3*e1*e2*e3+3*e3^2

M=matrix({{e1,e2,e3,i4}});
Phi=map(R,S,M);
ker Phi
-- also gilt
i4==e1*e2*e3-3*e3^2

M=matrix({{e1,e2,e3,i5}});
Phi=map(R,S,M);
ker Phi
-- also gilt
i5==e3^2

-- somit ist w gleich
-(2*(E1^3*E3-3*E1*E2*E3+3*E3^2)-(E1^2*E2^2-2*E2^3-2*E1^3*E3+4*E1*E2*E3-3*E3^2)+2*(E2^3-3*E1*E2*E3+3*E3^2)-2*(E1*E2*E3-3*E3^2)+6*(E3^2))

-- Test:
w==-(2*(e1^3*e3-3*e1*e2*e3+3*e3^2)-(e1^2*e2^2-2*e2^3-2*e1^3*e3+4*e1*e2*e3-3*e3^2)+2*(e2^3-3*e1*e2*e3+3*e3^2)-2*(e1*e2*e3-3*e3^2)+6*(e3^2))