vanishing points of a homogeneous ideal

ibahmani · Wed Jan 11, 2017 7:28 pm

I am trying to find vanishing points of a homogeneous ideal
I tried to use Singular to find but it seems there is not any function. Is there anyone who knows to do it? Has anyone tried it?

I=(wxy + x^2y + xy^2 + xyz, w^2y + wxy + wy^2 + wyz, w^2x + wx^2 + wxy + wxz, wxy)

Re: vanishing points of a homogeneous ideal

steenpass · Thu Jan 12, 2017 9:25 am

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