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