The zero set of this ideal is two-dimensional:
Code:
> ring r = 0, (w,x,y,z), dp;
> ideal I = wxy + x2y + xy2 + xyz, w2y + wxy + wy2 + wyz, w2x + wx2 + wxy + wxz, wxy;
> I = std(I);
> I;
I[1]=x2y+xy2+xyz
I[2]=wxy
I[3]=w2y+wy2+wyz
I[4]=w2x+wx2+wxz
> dim(I);
2
Given the above standard basis, it's relatively easy to compute the vanishing set 'by hand'. If not, then computing a primary decomposition might help:
Code:
> LIB "primdec.lib";
[snip]
> primdecGTZ(I);
[1]:
[1]:
_[1]=x
_[2]=w+y+z
[2]:
_[1]=x
_[2]=w+y+z
[2]:
[1]:
_[1]=y
_[2]=w+x+z
[2]:
_[1]=y
_[2]=w+x+z
[3]:
[1]:
_[1]=y
_[2]=x
[2]:
_[1]=y
_[2]=x
[4]:
[1]:
_[1]=x+y+z
_[2]=w
[2]:
_[1]=x+y+z
_[2]=w
[5]:
[1]:
_[1]=y
_[2]=w
[2]:
_[1]=y
_[2]=w
[6]:
[1]:
_[1]=x
_[2]=w
[2]:
_[1]=x
_[2]=w
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